Sagnac interferometer-type fiber-optic current sensor

ABSTRACT

In one embodiment, a Sagnac interferometer-type fiber-optic sensor includes a synchronous detection circuit to carry out synchronous detection of detected light signal with a phase modulation angular frequency of a phase modulator. A signal processing circuit calculates and outputs the magnitude of current to be measured using the signal detected in the synchronous detection circuit. A phase modulator driving circuit controls the driving of the phase modulator. The phase modulator driving circuit controls a phase modulation depth of the phase modulator so that the amplitude of the second-order harmonics and the fourth-order harmonics obtained by carrying out the synchronous detection of the detected light signal with the phase modulation angular frequency becomes the same.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based upon and claims the benefit of priority fromprior Japanese Patent Application No. 2009-281021, filed Dec. 10, 2009,the entire contents of which are incorporated herein by reference.

FIELD

The present invention relates to a Sagnac interferometer-typefiber-optic current sensor using optical phase modulation.

BACKGROUND

Various types of Sagnac interferometer-type fiber-optic current sensorsusing optical phase modulation are proposed. For example, “All-fiberSagnac current sensor” shown in FIG. 19 (refer to literature 1), the“Reciprocal reflection interferometer for a fiber-optic Faraday currentsensor” shown in FIG. 20 (refer to literature 2), “Production method fora sensor head for optical current sensor” shown in FIG. 21 (refer toNational Publication of translated version No. 2005-517961) and “Fiberoptic current sensor” shown in FIG. 22 (refer to National Publication oftranslated version No. 2002-529709) etc. are proposed.

Moreover, “The optical fiber current sensor and its calibrationequipment” shown in FIG. 23 (refer to Japanese Laid Open PatentApplication No. 2005-345350), and “Fiber optics apparatus and method foraccurate current sensing” shown in FIG. 24 (refer to NationalPublication of translated version No. 2000-515979), etc. propose methodsfor calculating an electrical current value from the detected lightsignal by a photodetector other than the conventional examples explainedabove.

According to the Sagnac interferometer-type fiber-optic current sensorsshown in FIGS. 19 to 24, the optical phase modulation with a fixedamplitude and a fixed angular frequency is provided to the light by anoptical phase modulator. As the optical phase modulator, a Pockels' cellphase modulator or a piezo-electric phase modulator configured bywinding an optical fiber around a cylindrical piezo-electric tubeelement is used. In addition, the above-mentioned angular frequency iscalled as a phase modulation angular frequency, and the above-mentionedamplitude is called as a phase modulation depth.

-   [Literature 1] G. Frosio, H. Hug, R. Dandliker, “All-fiber Sagnac    current sensor”, in Opto 92 (ESI Publications, Paris), p 560-564    (April, 1992)-   [Literature 2] G. Frosio and R. Dandliker, “Reciprocal reflection    interferometer for a fiber-optic Faraday current sensor”, Appl. Opt.    33, p 6111-6122 (September, 1994)

By the way, in the Sagnac interferometer-type fiber-optic currentsensors shown in above FIGS. 19-24, the Pockels' cell type optical phasemodulator or the piezo-electricity type phase modulator constituted bywinding an optical fiber around the cylinder piezo-electricity elementis used as the phase modulator. In either phase modulators, the light isphase-modulated by applying a voltage signal of the phase modulationangular frequency to the Pockels' cell element or the cylindricalpiezo-electric tube element. Since the phase modulation depth at thetime of the phase modulation is adjusted by the magnitude of theamplitude of the above-mentioned voltage signal, the phase modulationdepth actually applied to the light is dealt with as proportional to theamplitude of the voltage signal applied to the phase modulator.

However, the phase modulator has temperature characteristics, and thephase modulation efficiency also changes according to the phasemodulator's surrounding environmental temperature. Therefore, even ifthe amplitude of the voltage signal applied to the phase modulator iscontrolled at a set value, the phase modulation depth actually appliedto the light varies. Moreover, the change of the phase modulationefficiency by such phase modulators is also caused by the degradation ofthe phase modulator itself besides the temperature change.

As a result, since even if the phase modulator is driven with a fixedphase modulation depth, the phase modulation actually applied changes.Accordingly, the change of the modulation depth results in a variationof the sensing output of the current sensor. In both phase modulators ofthe Pockels' cell phase modulator and the piezo-electric tube phasemodulator as mentioned above, the light is phase-modulated by applying avoltage signal of the phase modulation angular frequency to the Pockels'cell element or the cylindrical piezo-electric tube element. Therefore,when a noise is overlapped on the voltage applied to the phasemodulator, the sensing output of the Sagnac interferometer-typefiber-optic current sensor is also changed similarly.

In such conventional modulation method, even if the magnitude of thephase modulation actually applied to the light changes, the modulationsystem is not equipped with a feedback system at all by which thefeedback operation is adjusted by detecting the change of the magnitude.

Furthermore, if a polarization extinction ratio in a propagation path oflight, especially between a phase modulator and a quarter-wave platechanges, it causes a problem that the sensing output of the Sagnacinterferometer-type fiber-optic current sensor changes, and themeasurement accuracy is also reduced. For example, in the Sagnacinterferometer-type fiber-optic current sensor shown in FIGS. 19 to 24,if the extinction ratio between the phase modulator and the quarter-waveplate deteriorates, the sensing output of the Sagnac interferometer-typefiber-optic current sensor changes. Accordingly, the measurementaccuracy reduces.

Moreover, in the Sagnac interferometer-type fiber-optic current sensorproposed in FIG. 19 and FIG. 20, it is assumed that the phase modulatoris optically connected with the quarter-wave plate by a polarizationmaintaining fiber. When mechanical stress, (the stress caused byvibration, sound, or a temperature change is included), is applied tothe polarization maintaining fiber, a crosstalk occurs between thelights which propagate along two optic axes of the polarizationmaintaining fiber, and the polarization extinction ratio changes.Therefore, the sensing output of the Sagnac interferometer-typefiber-optic current sensor also changes.

There are various factors by which the mechanical stress is applied tothe above polarization maintaining fiber. For example, people tramplethe polarization maintaining fiber, and stress may be applied to thefiber. Furthermore, when pulling out the polarization maintaining fiberfrom a case which hauses the quarter-wave plate and a sensing fiber, thepulled out portion of the polarization maintaining fiber from the caseis sealed by solder or bonding agent etc. in order to improve thesealing characteristics of the case. When the temperature change occursat the sealed portion, stress is applied to the polarization maintainingfiber due to the difference in the coefficient of thermal expansion ofthe materials.

Furthermore, when accommodating the polarization maintaining fiber tooptically connect between the phase modulator and the quarter-wave plateby winding in the shape of a coil, or when accommodating thepolarization maintaining fiber in a protective tube, the stress isapplied to the polarization maintaining fiber due to vibration oracoustic resonance. In this case, the polarization extinction ratio maychange, and the sensing output of the Sagnac interferometer-typefiber-optic current sensor may also change.

Moreover, when the phase modulator is optically connected with thequarter-wave plate by the polarization maintaining fiber using anoptical connector, the optical connector serves as mechanicalconnection. Accordingly, optic axis shift between the opticallyconnected polarization maintaining fibers results in a crosstalk and adecrease in the polarization extinction ratio. Even if the polarizationmaintaining fibers are ideally connected by the optical connector, sincethe optical connector serves as mechanical connector as above-mentioned,the vibration and temperature change are applied to the opticalconnector. Accordingly, the optical axis shift may occur between thepolarization maintaining fibers to be optically connected. Therefore, itis difficult to keep the polarization extinction ratio stable betweenthe phase modulator and the quarter-wave plate.

Furthermore, when the polarization maintaining fiber between the phasemodulator and the quarter-wave plate is optically connected by a fusionsplice method using electric discharge, a slight optical axis shift mayoccur between the polarization maintaining fibers similarly. Therefore,when the polarization maintaining fiber (light transmitting fiber) toconnect the sensor head having the quarter-wave plate with a signalprocessing unit having the phase modulator is separated once at a centerportion in the Sagnac interferometer-type fiber-optic current sensor,and the polarization maintaining fibers are optically connected again,the polarization extinction ratio may change. Therefore, the sensedoutput of the optical current sensor may change, that is, sensitivitychanges between before and after the separation of the above-mentionedpolarization maintaining fiber.

Thus, the deterioration of the polarization extinction ratio may takeplace in any portions of the polarizer, the phase modulator and thepolarization maintaining fiber to optically connect the opticalcomponents, and results in change of the sensing output of the opticalcurrent sensor.

Furthermore, when the signal processing methods proposed in FIG. 19,FIG. 21, FIG. 23, and FIG. 24 are used, and the polarization extinctionratio between the phase modulator and the quarter-wave plate changes,the sensed output of the Sagnac interferometer-type fiber-optic currentsensor changes. Therefore, the linearity of the input-and-outputcharacteristic deteriorates, and the measurement accuracy also falls dueto the change of the sensed output of the current sensor.

In addition, according to the signal processing method shown in FIG. 24,a system is adopted in which one optical phase difference of the twophase modulators is offset by the other phase modulator, and a currentvalue is calculated from the offset phase amount (practically, magnitudeof the voltage signal specifically applied to the phase modulator). Thedetection system is generally called a Serrodyne detection system. Inthis case, more than two phase modulators are required, and unless thephase modulation efficiency of two phase modulators is not the same, itis difficult to measure correctly the offset phase amount from thevoltage applied to the phase modulator. Therefore, the conventionalmethod results in problems that the measurement accuracy falls, the costis raised due to the use of two or more phase modulators, and thereliability of the sensor falls due to the increase in the number ofparts used.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate embodiments of the invention, andtogether with the general description given above and the detaileddescription of the embodiments given below, serve to explain theprinciples of the invention.

FIG. 1 is a figure showing a whole structure of Sagnacinterferometer-type fiber-optic current sensor according to a firstembodiment of the present invention.

FIG. 2 is a figure showing a structure of a Lyot type depolarizeraccording to the first embodiment of the present invention.

FIG. 3 is a figure showing a relation between the Bessel function andR=2δ sin ω_(m)α according to the first embodiment of the presentinvention.

FIG. 4 is a figure showing a relation between the waveform of a lightamount P_(out) and a peak value of the wave detected by a detectoraccording to the first embodiment of the present invention.

FIG. 5 is a figure showing a relation between the waveform of the lightamount P_(out) and the peak value of the wave detected with the detectorcorresponding to the magnitude of θ_(c) according to the firstembodiment of the present invention.

FIG. 6 is a figure showing a relation between a current correspondingoutput P_(k) and tan 4θ_(f) according to the first embodiment of thepresent invention. (in case |P_(2ω)|=|P_(4ω)|)

FIG. 7 is a figure showing a relation between θ_(c) and η(θ_(c))according to the first embodiment of the present invention. (in caseθ_(f)=0° and |P_(2ω)|=|P_(4ω)|)

FIG. 8 is a figure showing a relation between η and k′ according to thefirst embodiment of the present invention. (in case |P_(2ω)|=|P_(4ω)|)

FIG. 9 is a figure showing a relation between ω_(f) generated by themeasured current I and a ratio error according to the first embodimentof the present invention. (in case θ_(c)=0° and |P_(2ω)|=|P_(4ω)|)

FIG. 10 is a figure showing a relation between θ_(f) generated by themeasured current I and the ratio error according to the first embodimentof the present invention. (in case θ_(c)=3° and |P_(2ω)|=|P_(4ω)|)

FIG. 11 is a figure showing a relation between θ_(f) generated by themeasured current I and the ratio error according to the first embodimentof the present invention. (in case θ_(c)=6° and |P_(2ω)|=|P_(4ω)|)

FIG. 12 is a figure showing a relation between θ_(f) generated by themeasured current I and the ratio error according to the first embodimentof the present invention. (in case θ_(c)=10° and |P_(2ω)|=|P_(4ω)|)

FIG. 13 is a figure showing a relation between the current correspondingoutput P_(k) and tan 4θ_(f) according to a second embodiment accordingto the present invention. (in case |P_(1ω)|=0)

FIG. 14 is a figure showing the relation between θ_(c) and η(θ_(c))according to the second embodiment of the present invention. (in caseθ_(f)=0° and P_(1ω)=0).

FIG. 15 is a figure showing a relation between η and k′ according to thesecond embodiment of the present invention. (in case P_(1ω)=0)

FIG. 16 is a figure showing a relation between θ_(f) generated by themeasured current I and the ratio error according to the secondembodiment of the present invention. (in case θ_(c)=0° and P_(1ω)=0)

FIG. 17 is a figure showing a relation between θ_(f) generated by themeasured current I and the ratio error according to the secondembodiment of the present invention. (in case θ_(c)=10° and P_(1ω)=0)

FIG. 18 is a figure showing a whole Sagnac interferometer-typefiber-optic current sensor structure according to the second embodimentof the present invention.

FIG. 19 to FIG. 24 are figures showing the structures of theconventional current sensors, respectively.

DHILED DESCRIPTION OF THE INVENTION

A Sagnac interferometer-type fiber-optic current sensor according to anexemplary embodiment of the present invention will now be described withreference to the accompanying drawings wherein the same or likereference numerals designate the same or corresponding parts throughoutthe several views.

According to one embodiment, a Sagnac interferometer-type fiber-opticsensor includes: a synchronous detection circuit to carry outsynchronous detection of detected light signal with a phase modulationangular frequency of a phase modulator; a signal processing circuit tocalculate and output the magnitude of current to be measured using thesignal detected in the synchronous detection circuit; and a phasemodulator driving circuit to control the driving of the phase modulator;wherein the phase modulator driving circuit controls a phase modulationdepth of the phase modulator so that the amplitude of the second-orderharmonics and the fourth-order harmonics of the detected signal obtainedby carrying out the synchronous detection of the detected light signalwith the phase modulation angular frequency becomes the same.

According to other embodiment, a Sagnac interferometer-type fiber-opticsensor includes: a synchronous detection circuit to carry outsynchronous detection of the detected light signal with a phasemodulation angular frequency of a phase modulator; a signal processingcircuit to calculates and output the magnitude of current to be measuredusing the signal detected in the synchronous detection circuit; and aphase modulator driving circuit to control the driving of the phasemodulator; wherein the phase modulator driving circuit controls a phasemodulation depth so that the amplitude of the first-order harmonicsobtained by carrying out the synchronous detection of the detected lightsignal with the phase modulation angular frequency becomes “0”.

According to other embodiment, a Sagnac interferometer-type fiber-opticsensor includes: a synchronous detection circuit to carry outsynchronous detection of detected light signal with a phase modulationangular frequency of a phase modulator; a signal processing circuit tocalculate and output the magnitude of current to be measured using thesignal detected in the synchronous detection circuit; and a phasemodulator driving circuit to control the driving of the phase modulator;wherein the signal processing circuit includes; a normalization means tocalculate a reference value by dividing any one amplitude of theodd-order harmonics by any one amplitude of the even-order harmonics,where the harmonics are obtained by carrying out the synchronousdetection of the detected light signal with the phase modulation angularfrequency, and a compensation means to compensate the normalizedreference value with a ratio between any two amplitudes of the second-,fourth-, and sixth-order harmonics, and wherein the compensated value bythe compensating means is outputted as an output signal proportional tothe magnitude of the current to be measured.

1. First Embodiment

Next, the Sagnac interferometer-type fiber-optic current sensoraccording to the first embodiment of the present invention is explainedbelow with reference to FIGS. 1 to 12. Hereinafter, the basic structureof the optical current sensor according to the first embodiment andfundamental operation shall be explained, and the Jones matrix shall beused to explain the behavior of the light.

First, the basic structure of the Sagnac interferometer-type fiber-opticcurrent sensor according to the first embodiment is explained withreference to FIG. 1. FIG. 1 is a figure showing the basic structure ofthe Sagnac interferometer-type fiber-optic current sensor (Sagnacinterferometer-type optical CT).

As shown in FIG. 1, the Sagnac interferometer-type fiber-optic currentsensor includes a signal processing unit 100 which is composed of alight source driving circuit 101 described later, a light source 102, afiber coupler 103, an optical filter 104, a phase modulator 105, a delaycoil, a photodetector 106, a synchronous detection circuit 107 and asignal processing circuit 109.

Moreover, the current sensor includes a sensor head unit 300 which iscomposed of a quarter-wave plate 301 optically connected with the signalprocessing unit 100 through a light transmitting fiber 200, a sensingfiber 302, and a mirror 303.

In the signal processing unit 100, the light source driving circuit 101is a circuit to drive the light source 102. A small LED (light emittingdiode), a SLD (Super Luminescent Diode), etc. with coherentcharacteristics are used for the light source 102, and the output of thelight source is optically connected with the optical fiber. In thiscase, it is possible to use a single mode fiber or a polarizationmaintaining fiber as the optical fiber. The polarization maintainingfiber is used if it is required that the polarization maintainingcharacteristic is kept good.

In general, since the light which came out of the light source 102 canbe considered to be random light, the electric field component isdefined by the following equation 1.

$\begin{matrix}{E_{in} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 \\^{{({{\omega \; t} + \varphi_{r}})}}\end{pmatrix}}} & (1)\end{matrix}$

Here, the coefficient 1/√2 is provided to normalize so that an absolutevalue |E_(in)| of E_(in) becomes “1”. The phase φ_(r)(t) is a phasedifference which changes regardless of the x component of the E_(in),and the random light is expressed with φ_(r)(t).

The fiber coupler 103 is an optical branch means to branch the lightwhich comes out of the light source 102. One branch of the fiber coupler103 is optically connected with an optical filter 104 which forms thepolarizer. The fiber coupler 103 may be formed of either a single modefiber coupler or a polarization maintaining fiber coupler. Especially,if it is required to keep the polarization characteristics good, thepolarization maintaining fiber coupler is used.

The optical filter 104 is configured by the light polarizer 104 a. Thepolarization state of the actual light from the light source 102 may notbe uniform, although the light is random light. Therefore, a depolarizer104 b is inserted between the fiber coupler 103 and the polarizer 104 a.The optical filter 104 is composed of the depolarizer 104 b and thepolarizer 104 a. According to such structure, the output from thepolarizer 104 a is stabilized.

In detail, it is supposed that the polarization maintaining fiber isused for the light polarizer 104 a. Furthermore, a Lyot type depolarizerwhich is made by coupling the polarization maintaining fibers mutuallywith a rotation angle of approximately 45° between the optic axes of therespective polarization maintaining fibers is used as the depolarizer104 b as shown in FIG. 2. Here, the characteristic of the Lyot typedepolarizer 104 b is stabilized by setting the relation of the length ofthe respective polarization maintaining fibers connected mutually as 1:2or 2:1 (refer to JOURNAL Of Light Wave Technology Vol. LT1 No. 1 March1983 P71-P74).

Moreover, when the polarization maintaining fiber coupler is used as thefiber coupler 103 which is an optical branch means, the depolarizer 104b is constituted by coupling the lead portion (polarization maintainingfiber) of the polarization maintaining fiber coupler with thepolarization maintaining fiber (hereinafter expressed as a lead portionof the light polarizer) prepared for the purpose of propagating light byspecifying the optic axis without affecting to a function of thepolarizer 104 a with a rotation angle of approximately 45° between therespective optic axis. According to this structure, since it is notnecessary to insert the extra depolarizer, the number of coupling pointsto optically connect the fibers mutually also decreases. Furthermore, anoptical coupling loss, an optical transmission loss, and a lightpropagation polarization characteristics are more stabilized, and whichresults in economical structure.

In addition, also when using such depolarizer 104 b, a ratio of thelength of the lead portion (polarization maintaining fiber) of themaintaining fiber coupler 103 to the length of the lead portion(polarization maintaining fiber) of the polarizer 104 a is set to 1:2 or2:1. Thereby, the stability of the depolarizer 104 b is improved.

On the other hand, in the case where the polarizer 104 a uses thepolarization maintaining fiber, and the fiber coupler 103 uses a singlemode fiber coupler, a polarization maintaining fiber is inserted betweenthe lead portion (single mode fiber) of the single mode fiber coupler103 and the polarizer 104 a. The Lyot type depolarizer is constituted bycoupling the polarization maintaining fiber and the lead portion(polarization maintaining fiber) of the polarizer 104 a so that therespective optic axis of the polarization maintaining fibers rotate by45°. According to this structure, since it is not necessary to insert anextra depolarizer specially, the number of coupling points to opticallyconnect the fibers mutually also decreases. Furthermore, an opticalconnection loss, an optical transmission loss, and a light propagationpolarization characteristics are more stabilized, and which results ineconomical structure.

In addition, the stability of the depolarizer 104 b is raised by settingthe ratio of the length of the polarization maintaining fiber opticallyconnected to the single mode fiber coupler and the length of the leadportion (polarization maintaining fiber) of the polarizer 104 a to 1:2or 2:1.

Moreover, although the polarizer 104 a can be considered to be anelement which passes one of x and y components of the incident light, itis supposed that only x component of the incident light is passed, andthe Jones matrix L_(p) of the polarizer is defined as the followingequation 2 for explanation.

$\begin{matrix}{L_{p} = \begin{pmatrix}1 & 0 \\0 & 0\end{pmatrix}} & (2)\end{matrix}$

The polarizer 104 a is constituted by a fiber type polarizer using thepolarization maintaining fiber or a bulk element type fiber polarizerformed by combining a crystal element with the polarization maintainingfiber, etc. Here, the optic axes of the polarizer 104 a are specifiedwith the optic axis of the polarization maintaining fiber to constitutethe polarizer 104 a. That is, the polarizer 104 a shall be an elementwhich makes the light pass along one of the two optic axes of thepolarization maintaining fiber constituting the polarizer 104 a, and thepolarizer 104 a propagates only x component of the light.

A PZT type phase modulator constituted by winding the polarizationmaintaining fiber around a piezo-electric tube (PZT) or a Pockels' cellelement, etc. are used as the phase modulator 105.

When using the Pockels' cell element type phase modulator as the phasemodulator 105, the polarization maintaining fiber mentioned above isused for introducing the light to the Pockels' cell element andretrieving the light from the Pockels' cell element. In the phasemodulator 105, a relative phase modulation is provided to the lightwhich propagates one of two optic axes of the polarization maintainingfiber. That is, the direction of the phase modulation is defined by theoptic axis of the polarization maintaining fiber which constitutes thephase modulator 105.

Moreover, as the light propagating along the respective two optic axesof the polarization maintaining fiber, the light propagating in thephase modulator 105 without being affected by the phase modulation isused. In addition, the phase modulator 105 is supposed to provide arelative phase modulation to only x component of the light forexplanation.

Moreover, the optic axis of the polarization maintaining fiber whichconstitutes the phase modulator 105 is optically connected with thepolarization maintaining fiber which constitutes the polarizer 104 amentioned above in which the respective axes are rotated by 45°.Changing into the state where the respective polarization maintainingfibers are rotated by 45° becomes equivalent to make independent linearpolarization lights respectively pass along the two optic axes of thepolarization maintaining fiber which constitutes the phase modulator105. As a result, the Lyot type depolarizer is constituted by the leadportion (polarization maintaining fiber) of the polarizer 104 a and thelead portion of the phase modulator 105 (polarization maintainingfiber).

Here, when the polarization maintaining fiber (lead portion of the phasemodulator 105) optically connected to the polarizer 104 a (opticalfilter 104) side and prepared for the purpose of specifying an opticaxis and propagating the light to the phase modulator 105 withoutaffecting the function of the phase modulator 105 is optically connectedwith the polarization maintaining fiber (lead portion of the polarizer104 a) optically connected to the phase modulator 105 side prepared forthe purpose of specifying an optic axis and propagating the light topolarizer 104 a without affecting the function of the polarizer 104 a byrotating the respective axes by 45°, the ratio of the respective lengthof the polarization maintaining fibers is set to 1:2 or 2:1. Thereby,since it becomes possible to set the delay time lag between the x and ycomponents of the light propagating in the phase modulator 105 longerthan a possible interfering time (coherent time), the independency ofthe x and y components which propagate in the phase modulator 105becomes higher. Accordingly, the optical characteristic is morestabilized.

In addition, when using the Lyot type depolarizer for the optical filter104, the ratio of the full length of the Lyot type depolarizer used forthe optical filter 104 and the depolarizer constituted between thepolarizer 104 a and the phase modulator 105 is set to 1:2n or 2n:1(here, n is one or more integers). Since, interference by thepolarization components which remain in each depolarizer is alsocontrolled by this structure, an optical phase drift can be controlledas a result, and it becomes possible to control the zero point drift asthe optical current sensor.

Moreover, as mentioned above, the linear polarized light penetrates thephase modulator 105 after passing the light polarizer 104 a in the statewhere the optic axes of the polarizer 104 a and the optic axis of thephase modulator 105 are shifted by 45°. Thus, the behavior of the lightpropagating in the phase modulator 105 by rotating the optical axes by45° is expressed using the Jones matrix (equivalent to rotation by −45°with respect to incident direction angle) as the following equation 3.

$\begin{matrix}{L_{45{^\circ}} = {\begin{pmatrix}{\cos \; 45{^\circ}} & {{- \sin}\; 45{^\circ}} \\{\sin \; 45{^\circ}} & {\cos \; 45{^\circ}}\end{pmatrix} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & {- 1} \\1 & 1\end{pmatrix}}}} & (3)\end{matrix}$

In addition, the relative phase difference φ is provided to the xcomponent of the light propagating in the phase modulator 105. Here, ifthe light receives a phase modulation at time t=τ₁, the phase differenceφ is expressed with the following equation 4.

φ=δ sin ω_(m)(t−τ ₁)  (4)

where δ is a phase modulation depth and ω_(m) is a phase modulationangular frequency.

Therefore, the behavior of the light in the phase modulator 105 isexpressed with the following equation (5) using the Jones matrix.

$\begin{matrix}{L_{PM} = \begin{pmatrix}^{j\varphi} & 0 \\0 & 1\end{pmatrix}} & (5)\end{matrix}$

The two linearly polarized lights, which pass in the phase modulator 105so as to be restrained by each of the two optic axes of the polarizationmaintaining fiber forming the lead portion of the phase modulator 105,are led to the light transmitting fiber 200 formed of the polarizationmaintaining fiber. Then the two linearly polarized lights which passedthe light transmitting fiber 200 are introduced into the quarter-waveplate 301 of the sensor head unit 300.

The transmitting fiber 200 is optically connected with the quarter-waveplate 301 of the sensor head unit 300 to be mentioned later so that therespective optic axes rotate by 45° (equivalent to rotation of −45° withrespect to an incident direction angle). If this relation is expressedwith the Jones matrix, the matrix becomes the same as that of the aboveequation 3.

The sensor head unit 300 is constituted by the quarter-wave plate 301 togenerate a phase difference of ¼ wave, a sensing fiber 302 formed in theshape of a loop to generate the Faraday phase difference, and the mirror303 which reflects the light from the sensing fiber 302.

In the quarter-wave plate 301 of the sensor head unit 300, the phasedifference of ¼ wave (360°/4=90°) for the x component of the passinglight is generated. Consequently, the two linearly polarized lightspassed in the transmitting fiber 200 are converted into two circularpolarized lights of a clockwise propagated light and a counterclockwisepropagated light in opposite directions. The behavior of the light inthe quarter-wave plate 301 is expressed with the following Jones matrix(equation 6).

$\begin{matrix}{L_{QWP} = {\begin{pmatrix}^{\; 90{^\circ}} & 0 \\0 & 1\end{pmatrix} = \begin{pmatrix}i & 0 \\0 & 1\end{pmatrix}}} & (6)\end{matrix}$

The sensing fiber 302 is arranged so as to surround the current to bemeasured and generates the phase difference (Faraday phase difference)which is equivalent to the rotation according to the Faraday effect asshown in FIG. 1. The Faraday phase difference θ_(f) generated accordingto the Faraday effect in the sensing fiber 302 is expressed with afollowing equation 7.

θ_(f) =n·V·I  (7)

where,

-   -   n: the number of turns of the fiber which surrounds the current        to be measured,    -   V: the Verdet constant of the sensing fiber,    -   I: the current value to be measured.

In the above example, the case where the sensing fiber 302 is arrangedso as to surround the current is explained. However, the current coilmay surround the sensing fiber 302 reversely. In this case, n becomes anumber of turns of the current coil in which the current flows. TheFaraday phase difference in the sensing fiber 302 is expressed with theJones matrix as the following equation 8.

$\begin{matrix}{L_{F} = \begin{pmatrix}{\cos \; \theta_{f}} & {{- \sin}\; \theta_{f}} \\{\sin \; \theta_{f}} & {\cos \; \theta_{f}}\end{pmatrix}} & (8)\end{matrix}$

The mirror 303 is provided in one end of the sensing fiber 302, andreverses the y component of the light by reflecting two circularpolarized lights which passed in the sensing fiber 302 as shown inFIG. 1. The effect of the mirror 303 is expressed with Jones matrix asthe following equation 9.

$\begin{matrix}{L_{M} = \begin{pmatrix}1 & 0 \\0 & {- 1}\end{pmatrix}} & (9)\end{matrix}$

The two circular polarized lights reflected by the mirror 303 receivethe Faraday effect in the sensing fiber 302 again. However, since thedirection of the sensed magnetic field becomes opposite between thelight propagating to the mirror 303 direction and the light propagatingto the reflected direction by the mirror 303, the behavior of the lightpropagating in the inside of the sensing fiber 302 is expressed with thefollowing equation 10 using the Jones matrix.

$\begin{matrix}{L_{F^{\prime}} = \begin{pmatrix}{\cos \; \theta_{f}} & {\sin \; \theta_{f}} \\{{- \sin}\; \theta_{f}} & {\cos \; \theta_{f}}\end{pmatrix}} & (10)\end{matrix}$

The two circular polarized lights which passed in the sensing fiber 302are introduced into the quarter-wave plate 301, and generates the phasedifference of ¼ wave (360°/4=90°) only in the x component of the lightagain. Therefore, the behavior of the light in the quarter-wave plate isexpressed with the above equation 6 using the Jones matrix.

Then, the two rotated circular polarized lights in the oppositedirections by propagating in the sensing fiber 302 are respectivelyintroduced into the light transmitting fiber 200 (polarizationmaintaining fiber) as two linearly polarized lights after passing in thequarter-wave plate 301. In addition, the quarter-wave plate 301 andlight transmitting fiber 200 are optically connected in the state wherethe respective optic axes are arranged so as to rotate by 45° as theabove (equivalent to rotation of −45° with respect to an incidentdirection angle). Accordingly, the effect is expressed with the aboveequation 3 using the Jones matrix.

Here, when the linearly polarized light, which propagates along the xaxis of the two optic axes of the light transmitting fiber 200 andenters into the quarter-wave plate 301, re-enters into the lighttransmitting fiber 200 from the quarter-wave plate 301 via the sensingfiber 302, the light propagates along the y-axis of the two optic axesof the light transmitting fiber 200. Similarly, when the linearlypolarized light, which propagates along the y-axis of the two optic axesof the light transmitting fiber 200 and enters into the quarter-waveplate 301, re-enters into the light transmitting fiber 200 from thequarter-wave plate 301 via the sensing fiber 302, the light propagatesalong the x axis of the two optic axes of the light transmitting fiber200.

Then, while the light propagates in the light transmitting fiber 200again and enters into the phase modulator 105, the phase modulator 105provides a phase modulation with a relative phase difference φ′ to the xcomponent of the light which passes in the phase modulator 105.Supposing the light receives the phase modulation at time t=τ₂ here, thephase difference φ′ is expressed with the following equation 11.

φ′=δ sin ω_(m)(t−τ ₂)  (11)

If the behavior of the light in the phase modulator 105 in this case isexpressed with the Jones matrix, the behavior becomes as the equation12.

$\begin{matrix}{L_{{PM}^{\prime}} = \begin{pmatrix}^{{\varphi}^{\prime}} & 0 \\0 & 1\end{pmatrix}} & (12)\end{matrix}$

Further, the light which passed in the phase modulator 105 enters intothe optical filter 104. Since the phase modulator 105 and lightpolarizer are optically connected so that the respective axes arerotated by 45° (equivalent to rotating by −45° with respect to theincident direction angle), the effect is expressed with the Jones matrixas the equation (3).

Therefore, the two linearly polarized lights which propagate in thephase modulator 105 are synthesized and interferes each other afterpassing the point where the phase modulator 105 and the light polarizer105 are connected so that the respective optic axis of the phasemodulator 105 and light polarizer 105 are rotated by 45°. In addition,since the light polarizer 105 propagates only x component of theincident light, the Jones matrix L_(p) of the polarizer 105 is the sameas that of the above equation 2.

Then, the light which passed in the optical filter 104 is split into twolights after passing the fiber coupler 103, and one branched light isdetected with the photodetector 106. That is, one branched light of thetwo lights branched by the fiber coupler 103 is detected. Light/electricconverters (O/E conversion element), such as a photo-diode and aphotoelectron multiplier, are used as the photodetector 106. If theelectric field component of the light which enters into thephotodetector 106 is made into E_(out) here, the relation between E_(in)and E_(out) is expressed with the following equation 13.

$\begin{matrix}\begin{matrix}{E_{out} = {L_{p} \cdot L_{45{^\circ}} \cdot L_{{PM}^{\prime}} \cdot L_{45{^\circ}} \cdot L_{QWP} \cdot L_{F^{\prime}} \cdot L_{M} \cdot}} \\{{L_{F} \cdot L_{QWP} \cdot L_{45{^\circ}} \cdot L_{PM} \cdot L_{45{^\circ}} \cdot L_{p} \cdot E_{in}}} \\{= {\frac{1}{2\sqrt{2}}\begin{pmatrix}{{^{\varphi}^{{2\theta}_{t}}} + {^{{\varphi}^{\prime}}^{- {2\theta}_{t}}}} \\0\end{pmatrix}}}\end{matrix} & (13)\end{matrix}$

Since the detected light is measured with the light amount by thephotodetector 106, if the detected light amount is made into P_(out),the detected amount P_(out) can be expressed with the following equation14 because the light amount is proportional to the square of theelectric field.

$\begin{matrix}{{P_{out} \equiv {E_{out}}^{2}} = {\frac{1}{4}\left( {1 + {\cos \left( {\left( {\varphi - \varphi^{\prime}} \right) + {4\theta_{f}}} \right)}} \right)}} & (14)\end{matrix}$

Then, the detected light amount P_(out) by the photodetector 106 issynchronously detected with the phase modulation angular frequency ω_(m)using a synchronous detection circuit 107. Here, if the detected lightamount P_(out) is developed by the higher harmonics of the phasemodulation angular frequency ω_(m), P_(out) becomes as the followingequation 15.

$\begin{matrix}\begin{matrix}{P_{out} \equiv {E_{out}}^{2}} \\{= {\frac{1}{4}\left( {1 + {\cos \left( {\left( {\varphi - \varphi^{\prime}} \right) + {4\theta_{f}}} \right)}} \right)}} \\{= {\frac{1}{4}\left( {1 + {{\cos \left( {\varphi - \varphi^{\prime}} \right)}\cos \; 4\theta_{f}} - {{\sin \left( {\varphi - \varphi^{\prime}} \right)}\sin \; 4\theta_{f}}} \right)}} \\{= {\frac{1}{4}\begin{pmatrix}{1 + {{\cos \left( {2{\delta sin\omega}_{m}\alpha \; \cos \; \omega_{m}t_{0}} \right)}\cos \; 4\theta_{f}} -} \\{{\sin \left( {2{\delta sin}\; \omega_{m}\alpha \; \cos \; \omega_{m}t_{0}} \right)}\sin \; 4\theta_{f}}\end{pmatrix}}} \\{= {{\frac{1}{4}\left( {1 + {\left( {{J_{0}(R)} + {2{\sum\limits_{n = 1}^{\infty}{\left( {- 1} \right)^{n}{J_{2n}(R)}\cos \; 2n\; \omega_{m}t_{0}}}}} \right)\cos \; 4\theta_{f}}} \right)} -}} \\{{\frac{1}{4}\left( {2{\sum\limits_{n = 0}^{x}{\left( {- 1} \right)^{n}{J_{{2n} + 1}(R)}{\cos \left( {{2n} + 1} \right)}\omega_{m}t_{0}}}} \right)\sin \; 4\theta_{f}}} \\{= {P_{0\omega} + {P_{1\omega}\cos \; \omega_{m}t_{0}} + {P_{2\omega}\cos \; 2\omega_{m}t_{0}} + {P_{3\omega}\cos \; 3\omega_{m}t_{0}} +}} \\{{{P_{4\; \omega}\cos \; 4\omega_{m}t_{0}} + {P_{5\omega}\cos \; 5\; \omega_{m}t_{0}} + {P_{6\omega}\cos \; 6\omega_{m}t_{0}} + \ldots}}\end{matrix} & (15)\end{matrix}$

P_(0ω), P_(1ω), P_(2ω), P_(3ω), P_(4ω), P_(5ω), and P_(6ω) show theamplitude of the high harmonics, that is, zero-order harmonics,first-order harmonics, second-order harmonics, third-order harmonics,fourth-order harmonics, fifth-order harmonics and sixth-order harmonics,respectively when the synchronous detection of P_(out) is carried outwith the modulation angular frequency ω_(m), and the respectiveamplitudes become as the following equations 16 to 22.

$\begin{matrix}{P_{0\omega} = {\frac{1}{4}\left( {1 + {{J_{0}(R)}\cos \; 4\theta_{f}}} \right)}} & (16) \\{P_{1\omega} = {{- \frac{1}{2}}{J_{1}(R)}\sin \; 4\; \theta_{f}}} & (17) \\{P_{2\; \omega} = {{- \frac{1}{2}}{J_{2}(R)}\cos \; 4\theta_{f}}} & (18) \\{P_{3\; \omega} = {{+ \frac{1}{2}}{J_{3}(R)}\sin \; 4\theta_{f}}} & (19) \\{P_{4\; \omega} = {{+ \frac{1}{2}}{J_{4}(R)}\cos \; 4\theta_{f}}} & (20) \\{P_{5\; \omega} = {{- \frac{1}{2}}{J_{5}(R)}\sin \; 4\theta_{f}}} & (21) \\{P_{6\; \omega} = {{- \frac{1}{2}}{J_{6}(R)}\cos \; 4\theta_{f}}} & (22)\end{matrix}$

where, J_(n) is the n-th Bessel function and R=2δ sin ω_(m)α isreplaced.

Here, in the calculation, t₀=(t−(τ₂+τ₁)/2), α=(τ₂−τ₁)/2 are put, and thefollowing equations 23 and 24 are used.

$\begin{matrix}{{\cos \left( {R\; \cos \; \varphi} \right)} = {{J_{0}(R)} + {2{\sum\limits_{n = 1}^{\infty}\; {\left( {- 1} \right)^{n}{J_{2n}(R)}\cos \; 2\; n\; \varphi}}}}} & (23) \\{{\sin \left( {R\; \cos \; \varphi} \right)} = {2{\sum\limits_{n = 0}^{\infty}\; {\left( {- 1} \right)^{n}{J_{{2n} + 1}(R)}{\cos \left( {{2n} + 1} \right)}\varphi}}}} & (24)\end{matrix}$

α can be expressed with the following equation 25 using a light pathlength L_(opt) specified by time lag in which the two circular polarizedlights propagating in the fiber, that is, the Faraday element, receivethe phase modulation by the phase modulator.

$\begin{matrix}{\alpha = {\frac{\tau_{2} - \tau_{1}}{2} = {\frac{1}{2} \cdot \frac{L_{opt}}{c}}}} & (25)\end{matrix}$

In the case of the Sagnac interferometer-type fiber-optic current sensorshown in FIG. 1, the light propagation length from the modulationproviding point of the phase modulator 105 to the mirror is expressed asL, and the refractive index of the fiber is expressed as n_(s), theequation L_(opt)=2·n_(s)·L can be obtained. Here, c is the velocity oflight. Therefore, α is a system parameter which is decided uniquely bydetermining the light path length.

Next, the phase modulator driving circuit 108 feedback controls thephase modulator so that the absolute values of the amplitude of thesecond-order harmonics and the fourth-order harmonics which weresynchronously detected with the phase modulation angular frequency ω_(m)by the synchronous detection circuit 107 become the same, that is,|P_(2ω)|=|P_(4ω)|. The feedback control is carried out so that |P_(2ω)|becomes equal to |P_(4ω)| by adjusting a modulation depth δ. The signalprocessing circuit 109 uses the amplitude P_(3ω) of the third-orderharmonics as an output corresponding to the measured current.

Here, in the phase modulator driving circuit 108, the reason why thefeedback control is carried out so as to achieve |P_(2ω)|=|P_(4ω)| is asfollows. As shown in FIG. 3, when the values of the respective secondand fourth Bessel functions are the same among the Bessel functions ofthe second, third and fourth Bessel functions, the third Bessel functionbecomes the maximum. Namely, when |P_(2ω)|=|P_(4ω)|, |P_(3ω)| becomesthe maximum. Accordingly, a stable output without being easily affectedby influence of noise can be obtained.

Moreover, even if the modulation efficiency of the phase modulatorchanges, the actually applied phase modulation can make constant bycontrolling the phase modulation depth so that the respective amplitudesof the second-order harmonics P_(2ω) and the fourth-order harmonicsP_(4ω) become the same.

Here, when the emitted light amount from the light source or the lighttransmission loss in an optical circuit system changes, the light amountdetected with the photodetector 106 usually changes. Therefore, anormalization means 109 a of the signal processing circuit 109normalizes P_(3ω) with a peak wave value of the light amount P_(out)detected with the photodetector 106 as shown in FIG. 6 in order toobtain the output independent of the detected light amount.

There is other method to normalize P_(3ω) using the normalizing means109 a, that is, to normalize P_(3ω) by averaging the light amountP_(out) detected with the photodetector 106 by passing the electricsignal of P_(out) in a low-pass filter with a suitable time constant. Inthis case, if a visibility of an optical system is not good, and muchlight from an optical element returns, a DC noise light is overlapped onthe detected light amount. Therefore, the noise light causes an error atthe time of normalizing. As a consequence, the above method is notsuitable. However, it is also possible to normalize P_(3ω) with theaveraged value of a half-wave rectified electrical signal of thedetected light amount P_(out) by passing the electrical signal of thedetected light amount P_(out) in a DC cut filter (high pass filter) withsuitable time constant in order to remove the DC component of thedetected light amount P_(out).

The magnitude of P_(3ω) normalized by the signal of the detected lightamount P_(out) in this way becomes an output proportional to sin 4θ_(f)as the equation 19. In the signal processing circuit 109, the P_(3ω) isoutputted as a current corresponding output after carrying out anarcsine compensation to the normalized P_(3ω).

Here, the normalization to the third-order harmonics P_(3ω) by thenormalizing means 109 a of the signal processing circuit 109 ispractically explained with reference to FIG. 5. Hereinafter, a practicalproblem for normalization, and a normalizing method by the normalizingmeans 109 a of the signal processing circuit 109 are described in detailin order to solve the problem.

First, the sensor head unit 300 and the signal processing unit 100 arepractically separated at point A of the light transmitting fiber 200 asshown in FIG. 1 and are again optically connected at the same point A.This is because the locations where the sensor head unit 300 and thesignal processing unit 100 are installed are generally apart from eachother, and in the state where the sensor head unit 300 and the signalprocessing unit 100 are optically connected by the light transmittingfiber 200 without being separated, the construction efficiency is notgood.

Therefore, convenience is secured as follows. First, the sensor headunit 300 and the signal processing unit 100 are separated at point A inFIG. 1. After the respective installments completed, the separated lighttransmitting fibers (polarization maintaining fibers) 200 are opticallyconnected again by a fusion splice method or a light connector.

However, in the above connection method, the out of alignment of therespective optic axis of the light transmitting fibers 200 (polarizationmaintaining fibers) at the time of coupling causes a problem. In thefusion splice method, since the out of alignment of the respectiveoptical axis of the light transmitting fiber 200 (polarizationmaintaining fiber) is comparatively small, the misalignment is gettingto be controlled. However, the fusion splice method requires a fusionsplice machine for exclusive use, and still a small axial misalignmentof optic axis remains in fact.

Under such circumstances, a Sagnac interferometer-type fiber-opticcurrent sensor using the optical connector is desired in which theconnection between the mutual polarization maintaining fibers iscomparatively easy, and even if the out of the alignment of optic axesof the light connector is generated, the output is stable and a highaccuracy measurement is achieved.

Here, as for the general optical connector for the polarizationmaintaining fibers, the out of alignment of the optic axes of about ±1°is generated even in a excellent connector because the connection ismechanical. The optical misalignment of the optic axis is produced atthe time of connection by the optical connector. Moreover, also whentemperature changes, or vibration, etc. are applied to the opticalconnector, the optical misalignment is generated due to the mechanicalconnection.

Here, when the axial misalignment of the light transmitting fibers 200(polarization maintaining fibers) is generated, a crosstalk occursbetween the lights which propagate along two optic axes of thepolarization maintaining fibers, which results in a decrease in apolarization extinction ratio. Consequently, it is apparent that thesensitivity of the optical current sensor changes. Therefore, inconstituting a highly precise optical current sensor, difficulty hasbeen followed when the optical connector is used to optically connectthe light transmitting fiber 200 (polarization maintaining fiber) untilnow.

Hereinafter, assuming that the axial misalignment of the optical axistakes place between the polarization maintaining fibers at point A inFIG. 1, the result of having analyzed the behavior of the light isexplained below.

It supposed that the polarization maintaining fibers are opticallyconnected by the optical connector especially at point A in FIG. 1, andthe axial misalignment of only angle θ_(c) of the optical axis in thepolarization maintaining fibers is generated at the connecting point A(equivalent to an optical axis rotation by only angle −θ_(c) at thepoint A because the polarization maintaining fibers are connected bymaking the optical axes rotate by θ_(c) with respect to an incidentdirection angle). In this case, the Jones matrix showing the behavior ofthe light at point A is expressed with the following equation 26.

$\begin{matrix}{L_{A} = \begin{pmatrix}{\cos \; \theta_{c}} & {{- \sin}\; \theta_{c}} \\{\sin \; \theta_{c}} & {\cos \; \theta_{c}}\end{pmatrix}} & (26)\end{matrix}$

Here, the change of θ_(c) is equivalent to that of the polarizationextinction ratio γ between the respective polarization maintainingfibers, that is, between the phase modulator and the quarter-wave plate,and a relation γ=tan²θ_(c) (|θ_(c)|≦45°) is formed. Therefore, if θ_(c)increases, the polarization extinction ratio γ also increases, that is,ratio γ deteriorates.

As a consequence, when the electric field component of the light emittedfrom the light source is expressed with the above equation 1, theelectric field component E_(out) of the light which reaches to thephotodetector 106 is expressed with the following equation 27.

$\begin{matrix}{E_{out} = {\begin{matrix}{L_{p} \cdot L_{45{^\circ}} \cdot L_{{PM}^{\prime}} \cdot L_{A} \cdot L_{45{^\circ}} \cdot L_{QWP} \cdot} \\{L_{F^{\prime}} \cdot L_{M} \cdot L_{F} \cdot L_{QWP} \cdot L_{45{^\circ}} \cdot L_{A} \cdot L_{PM} \cdot L_{45{^\circ}} \cdot L_{p} \cdot E_{i\; n}}\end{matrix} = {\frac{1}{2\sqrt{2}}\begin{pmatrix}{{\left( {{^{\; \varphi}\cos \; \theta_{c}} - {\sin \; \theta_{c}}} \right){^{\; 2\; \theta_{f}}\left( {{^{{\varphi}^{\prime}}\sin \; \theta_{c}} + {\cos \; \theta_{c}}} \right)}} +} \\{\left( {{^{\; \varphi}\sin \; \theta_{c}} + {\cos \; \theta_{c}}} \right){^{{- }\; 2\; \theta_{f}}\left( {{^{\; \varphi^{\prime}}\cos \; \theta_{c}} - {\sin \; \theta_{c}}} \right)}} \\0\end{pmatrix}}}} & (27)\end{matrix}$

Therefore, the detection light amount P_(out) of the light detected withthe photodetector 106 becomes the following equation 28.

$\begin{matrix}{P_{out} = {{E_{out}}^{2} = {{\frac{1}{4}\left( {1 + {\cos \left( {\left( {\varphi - \varphi^{\prime}} \right) + {4\theta_{f}}} \right)}} \right)} + {\quad\left\lbrack \left. \quad {\frac{1}{4}\left( {{{- \cos^{2}}2\theta_{f}\sin^{2}2{\theta_{c}\left( {{\cos \left( {\varphi + \varphi^{\prime}} \right)} + {\cos \left( {\varphi - \varphi^{\prime}} \right)}} \right)}} + {2\; \sin \; 4\; \theta_{f}\sin^{2}\theta_{c}{\sin \left( {\varphi - \varphi^{\prime}} \right)}}} \right)} \right\rbrack \right.}}}} & (28)\end{matrix}$

The light amount P_(out) detected by the photodetector 106 issynchronously detected with the phase modulation angular frequency ω_(m)by the synchronous detection circuit 107 as mentioned above.Accordingly, the light amount P_(out) detected with the higher harmonicsof the phase modulation angular frequency ω_(m) is developed by thefollowing equation 29.

$\begin{matrix}{P_{out} = {{E_{out}}^{2} = {{{\frac{1}{4}\left( {1 + {\cos \left( {\left( {\varphi - \varphi^{\prime}} \right) + {4\theta_{f}}} \right)}} \right)} + \left\lbrack {\frac{1}{4}\left( {{{- \left( {{\cos \left( {\varphi + \varphi^{\prime}} \right)} + {\cos \left( {\varphi - \varphi^{\prime}} \right)}} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} + {2\mspace{11mu} {\sin \left( {\varphi - \varphi^{\prime}} \right)}\sin^{2}\theta_{c}\sin \; 4\theta_{f}}} \right)} \right\rbrack} = {{\frac{1}{4}\left( {1 + {{\cos \left( {\varphi - \varphi^{\prime}} \right)}\cos \; 4\theta_{f}} - {{\sin \left( {\varphi - \varphi^{\prime}} \right)}\sin \; 4\theta_{f}}} \right)} + {\quad {\left\lbrack {\frac{1}{4}\left( {{- \left( {{\cos \left( {\varphi + \varphi^{\prime}} \right)} + {\cos \left( {\varphi - \varphi^{\prime}} \right)}} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right)} \right\rbrack + {\quad {\left\lbrack {\frac{1}{2}{\sin \left( {\varphi - \varphi^{\prime}} \right)}\sin^{2}\theta_{c}\sin \; 4\theta_{f}} \right\rbrack = {{\frac{1}{4}\left( {1 + {{\cos \left( {2\delta \; \sin \; \omega_{m}{\alpha cos}\; \omega_{m}t_{0}} \right)}\cos \; 4\theta_{f}} - {{\sin \left( {2\delta \; \sin \; \omega_{m}\alpha \; \cos \; \omega_{m}t_{0}} \right)}\sin \; 4\; \theta_{f}}} \right)} + {\quad {\left\lbrack {\frac{1}{4}\left( {{- \left( {{\cos \left( {2\delta \; \cos \; \omega_{m}{\alpha sin}\; \omega_{m}t_{0}} \right)} + {\cos \left( {2\delta \; \sin \; \omega_{m}\alpha \; \cos \; \omega_{m}t_{0}} \right)}} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right)} \right\rbrack + {\quad{\left\lbrack {\frac{1}{2}{\sin \left( {2\delta \; \sin \; \omega_{m}\alpha \; \cos \; \omega_{m}t_{0}} \right)}\sin^{2}\theta_{c}\sin \; 4\; \theta_{f}} \right\rbrack = {{\frac{1}{4}\left( {1 + {\left( {{J_{0}(R)} + {2{\sum\limits_{n = 1}^{x}\; {\left( {- 1} \right)^{n}{J_{2n}(R)}\cos \; 2\; n\; \omega_{m}t_{0}}}}} \right)\cos \; 4\; \theta_{f}}} \right)} - {\frac{1}{4}\left( {{2{\sum\limits_{n = 0}^{\infty}\; {\left( {- 1} \right)^{n}{J_{{2n} + 1}\left( {\left. \quad R \right){\cos \left( {{2n} + 1} \right)}\omega_{m}t_{0}} \right)}\sin \; 4\; \theta_{f}}}} + {\quad{\left\lbrack {{- \frac{1}{4}}\left( {{J_{0}\left( R^{\prime} \right)} + {2{\sum\limits_{n = 1}^{\infty}\; {{J_{2n}\left( R^{\prime} \right)}\cos \; 2n\; \omega_{m}t_{0}}}}} \right)\sin^{2}2\; \theta_{c}\cos^{2}2\theta_{f}} \right\rbrack + \left\lbrack {{- \frac{1}{4}}\left( {{J_{0}(R)} + {2{\sum\limits_{n = 1}^{\infty}\; {\left( {- 1} \right)^{n}{J_{2n}(R)}\cos \; 2\; n\; \omega_{m}t_{0}}}}} \right)\sin^{2}2\theta_{c}\cos^{2}2\; \theta_{f}} \right\rbrack + {\quad{\left\lbrack {\frac{1}{2}\left( {2{\sum\limits_{n = 0}^{\infty}\; {\left( {- 1} \right)^{n}{J_{{2n} + 1}(R)}{\cos \left( {{2n} + 1} \right)}\omega_{m}t_{0}}}} \right)\sin^{2}\theta_{c}\sin \; 4\theta_{f}} \right\rbrack = {P_{0\omega} + {P_{1\omega}\cos \; \omega_{m}t_{0}} + {P_{2\; \omega}\cos \; 2\; \omega_{m}t_{0}} + {P_{3\omega}\cos \; 3\; \omega_{m}t_{0}} + {P_{4\; \omega}\cos \; 4\; \omega_{m}t_{0}} + {P_{5\; \omega}\cos \; 5\; \omega_{m}t_{0}} + {P_{6\; \omega}\cos \; 6\; \omega_{m}t_{0}} + \ldots}}}}}} \right.}}}}}}}}}}}}}}} & (29)\end{matrix}$

Here, P_(0ω), P_(1ω), P_(2ω), P_(3ω), P_(4ω), P_(5ω), and P_(6ω)respectively show the amplitudes of the zero-order harmonics,first-order harmonics, second-order harmonics, third-order harmonics,fourth-order harmonics, fifth-order harmonics, and sixth-order harmonicsat the time of having carried out the synchronous detection of theP_(out) with the modulation angular frequency ω_(m). They are expressedwith the following equations 39 to 36, respectively.

$\begin{matrix}{P_{0\omega} = {{\frac{1}{4}\left( {1 + {{J_{0}(R)}\cos \; 4\; \theta_{f}}} \right)} + \left\lbrack {{- \frac{1}{8}}\left( {{J_{0}(R)} + {J_{0}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right\rbrack}} & (30) \\{P_{1\omega} = {{{- \frac{1}{2}}{J_{1}(R)}\sin \; 4\; \theta_{f}} + \left\lbrack {{+ \frac{1}{2}}{J_{1}(R)}\sin^{2}\theta_{c}\sin \; 4\; \theta_{f}} \right\rbrack}} & (31) \\{P_{2\omega} = {{{- \frac{1}{2}}{J_{2}(R)}\cos \; 4\theta_{f}} + \left\lbrack {{+ \frac{1}{4}}\left( {{J_{2}(R)} - {J_{2}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right\rbrack}} & (32) \\{P_{3\omega} = {{{+ \frac{1}{2}}{J_{3}(R)}\sin \; 4\theta_{f}} + \left\lbrack {{- \frac{1}{2}}{J_{3}(R)}\sin^{2}\theta_{c}\sin \; 4\; \theta_{f}} \right\rbrack}} & (33) \\{P_{4\omega} = {{{+ \frac{1}{2}}{J_{4}(R)}\cos \; 4\; \theta_{f}} + \left\lbrack {{- \frac{1}{4}}\left( {{J_{4}(R)} + {J_{4}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right\rbrack}} & (34) \\{P_{5\omega} = {{{- \frac{1}{2}}{J_{5}(R)}\sin \; 4\theta_{f}} + \left\lbrack {{+ \frac{1}{2}}{J_{5}(R)}\sin^{2}\theta_{c}\sin \; 4\theta_{f}} \right\rbrack}} & (35) \\{P_{6\omega} = {{{- \frac{1}{2}}{J_{6}(R)}\cos \; 4\; \theta_{f}} + \left\lbrack {{+ \frac{1}{4}}\left( {{J_{6}(R)} - {J_{6}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}} \right\rbrack}} & (36)\end{matrix}$

where J_(n) is the n-th Bessel function, and the relations R=2δ sinω_(m)α and R′=2δ cos ω_(m)α are put.

In addition, in the above calculation, the relation t₀=t−(τ₂+τ₁)/2, andα=(τ₂−τ₁)/2 are put. Furthermore, in addition to the above equations 23and 24, the relation of the following equation 37 is used.

$\begin{matrix}{{\cos \left( {R\mspace{11mu} \sin \; \varphi} \right)} = {{J_{0}(R)} + {2{\sum\limits_{n = 1}^{\infty}\; {{J_{2n}(R)}\cos \; 2n\; \varphi}}}}} & (37)\end{matrix}$

In an ideal case of θ_(c)=0, that is, when there is no out of alignmentbetween the optic axis of the polarization maintaining fibers, theamplitudes of the harmonics become the formula in which the portion of [] in the equations 28 to 36 is removed, that is, the same as theequations 14 to 22.

In the Sagnac interferometer-type fiber-optic current sensor accordingto this embodiment as mentioned above, the feedback control is carriedout so that |P_(2ω)|=|P_(4ω)|, and the amplitude of the third-orderharmonics P_(3ω) is used as an output corresponding to the measuredcurrent.

However, in case θ_(c) is large, for example, θ_(c)=10°, the waveform ofP_(out) calculated by the equation 29 is distorted as shown in FIG. 5according to the above analysis results. Therefore, the normalizationerror of P_(3ω) becomes large if using the peak value of P_(3ω) or thehalf-wave rectified value by removing the DC component from the detectedlight amount P_(out). As a result, even if an arcsine compensation iscarried out to the normalized P_(3ω), the problem that the linearitywith θ_(f), i.e., the measurement current I deteriorates is caused.

Furthermore, since the amplitude of the third-order harmonics P_(3ω)depends on the axial misalignment angle θ_(c) of the optic axis of therespective polarization maintaining fibers as shown in the equation 33,the output of the optical current sensor changes with ω_(c). That is, ifthe above is summarized, the following two points are raised asproblems.

(1) Since the normalization of P_(3ω) using P_(out) is accompanied withthe distortion of the waveform of P_(out), the linearity with θ_(f)becomes worse.(2) The output of the optical current sensor changes with the magnitudeof θ_(c).

Then, in this embodiment, the normalization means 109 a of the signalprocessing circuit 109 normalizes P_(3ω) by dividing the P_(3ω) with oneof |P_(2ω)|, |P_(4ω)|, and |P_(6ω)|, or sum of |P_(2ω)| and |P_(4ω)| inorder to solve above problems. In the Sagnac interference type opticalcurrent sensor according to the first embodiment, since the P_(3ω) isfeedback controlled so as to |P_(2ω)|=|P_(4ω)|, the normalization ofP_(3ω) with the |P_(2ω)| is equivalent to the normalization with|P_(4ω)|.

Moreover, the normalization of |P_(3ω)| with the sum of |P_(2ω)| and|P_(4ω)| is also substantially equivalent to the normalization with|P_(2ω)| or |P_(4ω)|. The normalization method using the sum of |P_(2ω)|and |P_(4ω)| is more suitable normalization method at the time of thefeedback control to average the slight difference between |P_(2ω)| and|P_(4ω)|. In addition, since the normalization of P_(3ω) with the sum of|P_(2ω)| and |P_(4ω)| is equivalent to the normalization with |P_(2ω)|or |P_(4ω)|, the explanation is omitted.

Hereinafter, the reason why the normalization means 109 a of the signalprocessing circuit 109 normalizes P_(3ω) by dividing the P_(3ω) with|P_(2ω)|, |P_(4ω)|, or |P_(6ω)|, or sum of |P_(2ω)| and |P_(4ω)|.

First, in the case the current is not flowing, that is, θ_(f)=0, theodd-order harmonics at the time of carrying out the synchronousdetection of the light detected with the photodetector 106 with themodulation angular frequency by the synchronous detection circuit 107becomes “0”, and only the even-order harmonics are detected as apparentfrom above equations 29 to 36. Therefore, the even-order harmonics arethe signals reflecting the magnitude of the detected light amount.Furthermore, since P_(2ω) and P_(4ω) become the harmonics with thelargest absolute values of the amplitude except for the zero-orderharmonics among the even-order harmonics, P_(2ω) and P_(4ω) are signalseffective in the normalization of P_(3ω) as compared with the highereven-order harmonics,

In addition, since the influenced extent (change rate) of P_(2ω) andP_(4ω) with respect to θ_(c) is substantially the same as that ofP_(3ω), the normalization of P_(3ω) with |P_(2ω)| or |P_(4ω)| is morestable to the change of θ_(c). Further, the change of the detected lightamount accompanied with the change of the optical transmission loss ofthe optical system or the emitted light amount of the light source actsat the same rate to all the components of the harmonics signals to whichthe synchronous detection was carried out with the modulation angularfrequency. Accordingly, the normalization value acquired by theabove-mentioned normalization method is a value which is not affected bythe influence of the change of the detected light amount.

On the other hand, although the zero-order harmonics is also a valuereflecting the detected light amount, the harmonics becomes DC like andresults in an error at the time of normalizing because the DC like noiselight is overlapped on the detected light amount when the visibility ofthe optical system is not good, and there is much return light from theoptical element. Accordingly, the zero-order harmonics is not suitable.

Moreover, sixth-order harmonics P_(6ω) is a signal effective innormalization of P_(3ω), and can be used. However, the magnitude of theeven-order harmonics more than eighth-order is smaller as compared withthe second-, fourth-, and sixth-order harmonics and tends to be affectedby the influence of the noise. In addition, when the polarizationextinction ratio change is large and the value of the current to bemeasured is also large, the polarity may be inverted. Accordingly, theeven-order harmonics more than eighth-order are not suitable. Inaddition, although the sixth-order harmonics P_(6ω) is a signaleffective in normalization of P_(3ω), it supposes that P_(6ω) isconsidered to be treated as the same way as the second-order harmonicsP_(2ω) and the fourth-order harmonics P_(4ω). Accordingly theexplanation about P_(6ω) is omitted.

Next, the embodiment in which the compensation means 109 b of the signalprocessing circuit 109 carries out the arctangent compensation to thenormalized output of harmonics as mentioned above is explained belowwith reference to FIGS. 6 to 8.

First, in the ideal case, that is, θ_(c)=0, the normalized output P_(3ω)with |P_(2ω)| or |P_(4ω)| becomes an output proportional to tan 4θ_(f).Therefore, the compensation means 109 b of the signal processing circuit109 carries out the arctangent compensation to the output proportionalto tan 4θ_(f). Accordingly, even in the case of the large currentmeasurement under the condition that θ_(f) is large, it is possible toobtain a highly precise output.

Since the light detected with the photodetector 106 specifically followsthe above equations 15 to 22 in the case of θ_(c)=0, the currentcorresponding output P_(3ω)′ which is obtained by normalizing P_(3ω)with |P_(2ω)| or |P_(4ω)| under the condition |P_(2ω)|=|P_(4ω)|, isexpressed with the following equations 38 and 39.

$\begin{matrix}{P_{3\omega}^{\prime} = {\frac{P_{3\omega}}{P_{2\omega}} = \frac{P_{3\omega}}{P_{4\omega}}}} & (38) \\{= {{k \cdot \tan}\; \theta_{f}}} & (39)\end{matrix}$

where k is expressed with the following equation 40.

$\begin{matrix}{k = {\frac{J_{3}(R)}{{J_{2}(R)}} = {\frac{J_{3}(R)}{{J_{4}(R)}}\mspace{11mu} \left( {\approx 1.4} \right)}}} & (40)\end{matrix}$

According to the equation 40, the constant k does not depend on θ_(f),but is uniquely decided under the condition |P_(2ω)|=|P_(4ω)|, that is,k≈1.4 (refer to FIG. 3). Therefore, if P_(k) is replaced by thefollowing equations 41 and 42, the following equations 43 and 44 areobtained.

$\begin{matrix}{P_{k} = \frac{P_{3\omega}^{\prime}}{k}} & (41) \\{\approx \frac{P_{3\omega}^{\prime}}{1.4}} & (42) \\{\theta_{f} = {{\frac{1}{4}{arc}\; \tan \; P_{k}} = {\frac{1}{4}{arc}\; \tan \frac{P_{3\omega}^{\prime}}{k}}}} & (43) \\{\approx {\frac{1}{4}{arc}\; \tan \frac{P_{3\omega}^{\prime}}{1.4}}} & (44)\end{matrix}$

According to above equations, the compensation means 109 b of the signalprocessing circuit 109 can obtain θ_(f), i.e., the output proportionalto the current I by carrying out the arctangent compensation to thecurrent corresponding output P_(k) which is obtained by dividing P_(3ω)′with the constant k (refer to above equation 7). In addition, sinceapproximation of θ_(f)≈P_(k)/4 is realized in the area where the currentI to be measured is small (θ_(f)<<1), the compensation by arctangent canimprove the measurement accuracy of the Sagnac interferometer-typefiber-optic current sensor in the area where the current I to bemeasured is large.

On the other hand, since the light detected with the photodetector 106specifically follows the above equations 29 to 36 in the case ofθ_(c)≠0, the current corresponding output P_(k) which is obtained bydividing P_(3ω)′ with the constant k≈1.4 under the condition|P_(2ω)|=|P_(4ω)| like the case of θ_(c)=0, is expressed with thefollowing equations 45 and 46.

$\begin{matrix}{P_{k} = {\frac{P_{3\omega}}{k{P_{2\omega}}} = \frac{{{J_{3}(R)}\sin \; 4\; \theta_{f}} - {2{J_{3}(R)}\sin^{2}\theta_{c}\sin \; 4\; \theta_{f}}}{k{\begin{matrix}{{{- {J_{2}(R)}}\cos \; 4\; \theta_{f}} +} \\\left( {{J_{2}(R)} - {{J_{2}\left( R^{\prime} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}}} \right)\end{matrix}}}}} & (45) \\{= {\frac{P_{3\omega}}{k{P_{4\omega}}} = \frac{{{J_{3}(R)}\sin \; 4\; \theta_{f}} - {2{J_{3}(R)}\sin^{2}\theta_{c}\sin \; 4\; \theta_{f}}}{\left. {k{\begin{matrix}{{{J_{4}(R)}\cos \; 4\; \theta_{f}} -} \\\left( {{J_{4}(R)} + {{J_{4}\left( R^{\prime} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}}} \right.\end{matrix}}} \right)}}} & (46)\end{matrix}$

Here, the relation between P_(k) and tan 4θ_(f) shows a good linearityas shown in FIG. 6 because the respective change ratios of P_(2ω),P_(4ω), and P_(3ω) with respect to θ_(c) are almost the same. Therefore,it is effective to use the normalized value which is obtained bynormalizing P_(3ω) with |P_(2ω)| or |P_(4ω)| as a current correspondingoutput to suppress deterioration of the linearity with θ_(f), i.e., themeasurement current.

However, as shown in FIG. 6, in the case of θ_(c)≠0, P_(k) changesdepending on the magnitude of θ_(c). Even if the compensation means 109b of the signal processing circuit 109 calculates the arctangent ofP_(k) obtained by the equations 45 and 46, an error is generated not alittle. In addition, P_(k) which is the value obtained by normalizingP_(3ω) with |P_(2ω)| or |P_(4ω)| shows a good linearity with tan 4θ_(f).Therefore, highly precise measurement is realizable by calibrating withthe current which satisfies the condition 4θ_(f)<<1 for every opticalcurrent sensor as far as θ_(c) is small or constant. However, suchcalibration method is not practical.

As shown in FIG. 6, although P_(k) slightly changes with respect to tan4θ_(f) depending on the magnitude of θ_(c), P_(k) is nearly proportionalto tan 4θ_(f), and tan 4θ_(f) and P_(k) are related as the followingequation 47 using a coefficient k′ (in an ideal case of θ_(c)=0 andk′=1).

tan 4θ_(f) =k′·P _(k)  (47)

Therefore, if the magnitude of θ_(c) and the change of k′ by θ_(c) areknown, tan 4θ_(f) is able to be calculated using the equation 47. Whenthe above equations 32 and 34 are compared with the equation 36, themagnitude of θ_(c) can be estimated using a value obtained by dividingthe |P_(2ω)| or |P_(4ω)| with |P_(6ω)| employing the compensation means109 b of the signal processing circuit 109. Here, the attention is paidonly about the even-order harmonics because the amplitude of theeven-order harmonics does not become “0” even in the case of θ_(f)=0.

In the above case, the value obtained by dividing the |P_(2ω)| or|P_(4ω)| with |P_(6ω)| is used as a cue to know the magnitude of θ_(c).However, other higher even-order harmonics can be also used in the sameway. But, the magnitudes of the even-order harmonics more thaneighth-order are small as compared with the second-, fourth-, andsixth-order harmonics, and they tend to be affected by the influence ofthe noise. In addition, when the change of the polarization extinctionratio or the crosstalk of the fiber is large and the value of thecurrent to be measured is also large, the even-order harmonics more thaneighth-order are not suitable because the polarity of the harmonics maybe inverted.

Therefore, a method to estimate θ_(c) using |P_(2ω)|, |P_(4ω)|, and|P_(6ω)| can raise the measuring accuracy more. If a value obtained bydividing |P_(2ω)| or |P_(4ω)| with |P_(6ω)| is defined as η, η isexpressed with the following equation 48.

$\begin{matrix}{\eta = {\frac{P_{2\omega}}{P_{6\; \omega}} = \frac{P_{4\omega}}{P_{6\; \omega}}}} & (48)\end{matrix}$

Here, η becomes a function of θ_(c) and θ_(f) as apparent from the aboveequations 32, 34, and 36 under the condition of |P_(2ω)|=|P_(4ω)|.However, if the value of θ_(c) is small or constant, η defined by theequation 48 hardly changes with respect to θ_(f) which satisfies4θ_(f)<<1. Therefore, η is approximately a function of θ_(c).

That is, in the case of θ_(f)=0, η is expressed with the followingequation 49 under the condition |P_(2ω)|=|P_(4ω)| from the aboveequations 32, 34, 36, and 48.

$\begin{matrix}{\eta = {\frac{P_{2\omega}}{P_{6\omega}} = \frac{{{- {J_{2}(R)}} + {\left( {{J_{2}(R)} - {J_{2}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}{{{- {J_{6}(R)}} + {\left( {{J_{6}(R)} - {J_{6}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}}} & (49) \\{= {\frac{P_{4\; \omega}}{P_{6\; \omega}} = \frac{{{J_{4}(R)} - {\left( {{J_{4}(R)} + {J_{4}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}{{{- {J_{6}(R)}} + {\left( {{J_{6}(R)} - {J_{6}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}}} & (50)\end{matrix}$

Thus, the relation between θ_(c) and η calculated by the equations 49and 50 is shown in FIG. 7. According to FIG. 7, η is decided uniquely byθ_(c) and it becomes possible to get to know the magnitude of θ_(c)approximately by measuring η. Thus, since θ_(c) can be approximatelyknown from η, a relation between η and k′ related to θ_(c) is estimated,k′ can be defined as follows from the above equation 47.

$\begin{matrix}{k^{\prime} = \frac{\tan \; 4\; \theta_{f}}{P_{k}}} & (51)\end{matrix}$

First, k′ for θ_(f) a constant value which is not “0” and satisfies4θ_(f)<<1 under condition of |P_(2ω)|=|P_(4ω)| is obtained using theequation 51, and η is also obtained by the equation 48. Consequently,the result of the estimation about the estimation between k′ and η isshown in FIG. 8. In this estimation, 4θ_(f)=1.0405E−05 (rad).

According to FIG. 8, k′ can be approximately obtained and uniquelydecided if η defined by the equation 48 is used, and it turns out thatthe compensation by the third-order compensation is enough with respectto η. The estimated k′ is expressed using η with the following equation52.

k′=2.578×10⁻³·η³−4.137×10⁻²·η²+2.352×10⁻¹·η+5.354×10⁻¹  (52)

If the θ_(f) is approximated under the condition |P_(2ω)|=|P_(4ω)|, theapproximated value becomes the following equation 53 by the equations 47and 52

$\begin{matrix}{\theta_{f} \approx {\frac{1}{4}{\arctan \left( {\left( {\sum\limits_{n = 0}^{3}{a_{n} \cdot \eta^{n}}} \right) \cdot P_{k}} \right)}}} & (53)\end{matrix}$

where, a_(n) is a constant, that is, the respective a₀ to a₃ are asfollows. a₀=5.354×10⁻¹, a₁=2.352×10⁻¹, a₂=−4.137×10⁻², anda₃=2.578×10⁻³.

Moreover, P_(k) is expressed with the following equation 54, and η isexpressed with above equation 48.

$\begin{matrix}{P_{k} = {{\frac{1}{1.4} \cdot \frac{P_{3\omega}}{P_{2\omega}}} = {\frac{1}{1.4} \cdot \frac{P_{3\omega}}{P_{4\omega}}}}} & (54)\end{matrix}$

Therefore, θ_(f) proportional to the measured current value I expectedto be outputted finally can be obtained by using four harmonics P_(2ω),P_(3ω), P_(4ω), and P_(6ω), based on the equation 53. Thereby, accordingto the equation 53, in the case where the misalignment angular value ofthe polarization maintaining fibers is small or constant, that is, thepolarization extinction ratio is small and constant, the approximationof θ_(f)≈P_(k)/4 is realized in the area where the current to bemeasured is small (θ_(f)<<1). Furthermore, the arctangent compensationis carried out in the area where the current I to be measured is large.Accordingly, the measurement accuracy of the Sagnac interferometer-typefiber-optic current sensor can be improved.

Next, in order to show how much precisely the approximation by the aboveequation 53 is performed, the relation between θ_(f) and the ratio errorwith respect to the misalignment angle θ_(c) of the polarizationmaintaining fibers at Point A of FIG. 1 is shown in FIGS. 9 to 12 bydefining the ratio error as the following equation 55.

$\begin{matrix}{{{Ratio}\mspace{14mu} {error}\mspace{14mu} (\%)} = {\left( \frac{\frac{1}{4}{\arctan \left( {\left( {\sum\limits_{n = 0}^{3}{a_{n} \cdot \eta^{n}}} \right) \cdot P_{k}} \right)}}{\theta_{f}} \right) \times 100}} & (55)\end{matrix}$

The approximate expression shown in the above equation 53 can realizehighly precise approximation as shown in FIGS. 9 to 12, and it becomespossible to approximate the current of θ_(f)<0.06 (rad) in the range ofθ_(c)=0° to 10° with accuracy less than 0.1%. That is, according to theequation 53, a highly precise current measurement is realizable bycompensating the output of the optical current sensor by thecompensation means 109 b.

For the reference, when a SLD with a wavelength of 830 nm is used forthe light source in FIG. 1, and a quartz fiber is used for the sensingfiber, the Verdet constant number V of the quartz is 2.6×10⁻⁶ (rad/A).Accordingly, if the number of turns n of the sensing fiber is made intoone, the current magnitude in the range of θ_(f)<0.06 (rad) correspondsto 23 kA or less.

According to the above first embodiment, the phase modulation depth ofthe second-order harmonics and the fourth-order harmonics obtained bycarrying out the synchronous detection of the light amount detected withthe photodetector 106 with the phase modulation angular frequency iscontrolled so that the respective amplitudes are set so as to be thesame. Then, a compensation is carried out to the normalized amplitude ofthe third-order harmonics with the second-order harmonics, thefourth-order harmonics, or the sum of the second-order harmonics and thefourth-order harmonics using a value obtained by dividing the amplitudeof the second-order harmonics or the fourth-order harmonics with theamplitude of the sixth-order harmonics. Further, the arctangentcompensation is carried out to the compensated third-order harmonics,and the output of the arctangent of the compensated third-orderharmonics is finally used as an output of the Sagnac interferometer-typefiber-optic current sensors. Accordingly, even if it is a case where thepolarization extinction ratio of the light transmitting fiber 200 whichconnects the sensor head unit 300 and the signal processing unit 100optically deteriorates, it becomes possible to offer the optical currentsensor in which a highly precise current measurement is possible.

Furthermore, according to this embodiment, since the amplitudes of thesecond-order harmonics and the fourth-order harmonics are controlled soas to become the same, the third-order harmonics can be stably made intothe maximum, and it becomes possible to make the third-order harmonicsrobust against the influence by a noise. Moreover, the phase modulationactually provided to the light can make constant by controlling thephase modulation depth so that the amplitudes of the second-orderharmonics and the fourth-order harmonics become the same, even if thephase efficiency of the phase modulator changes.

Moreover, even if the detected light amount changes with the emittedlight amount of the light source and optical transmission loss of theoptical propagation path, etc., the stable output proportional to themeasured current can be obtained by normalizing the third-orderharmonics with the even-order harmonics. Especially, since the values ofthe second-order harmonics, the fourth-order harmonics, and thesixth-order harmonics are large compared with the amplitude of otherhigh even-order harmonics, they cannot be easily affected by influenceof the noise. In addition, even if the extinction ratio between themodulator and the quarter-wave plate changes, the normalized outputusing one of the amplitudes of the second-order harmonics, thefourth-order harmonics, and the sixth-order harmonics, or the sum of theamplitudes of the second-order harmonics and the fourth-order harmonicsshows a good linearity with the measured current, if the value of thepolarization extinction ratio between the modulator and the quarter-waveplate is small or constant.

Moreover, in the process for controlling the second-order harmonics andthe fourth-order harmonics to the same value, the change direction ofthe second-order harmonics and the fourth-order harmonics is reverse.Accordingly, if the third-order harmonics is normalized with the sum ofthe amplitudes of the second-order harmonics and the fourth-orderharmonics, it becomes possible to perform the normalization more stablethan to perform the normalization of the third-order harmonics with thesecond-order harmonics or the fourth-order harmonics simply.

Furthermore, according to this embodiment, the signal processing circuitincludes a compensation means to compensate the reference value usingany two ratios between the amplitudes of the second-, fourth-, and thesixth-order harmonics, then outputs the compensated value by thecompensation means as a signal proportional to the magnitude of thecurrent to be measured.

According to above embodiment, even if the polarization extinction ratiobetween the phase modulator and the quarter-wave plate changes, it ispossible to obtain the normalized value with a small sensitivityvariation by compensating the normalized output using any two amplitudesof the second-, fourth- and sixth-order harmonics as mentioned above. Asa result, it becomes possible not only to acquire the linearity of theinput-output characteristic but also to control the sensitivity changeof the optical current sensor. Accordingly, a highly precise Sagnacinterferometer-type fiber-optic current sensor can be realized.

According to this embodiment, even if the polarization extinction ratiobetween the phase modulator and the quarter-wave plate changes, it ispossible to obtain the compensated normalized output value with a smallsensitivity variation by compensating the normalized reference valueobtained by normalizing the odd-order harmonics obtained by carrying outthe synchronous detection with any two of the second-, fourth- andsixth-order harmonics. As a result, it becomes possible not only toacquire the linearity of the input-output characteristic but also tocontrol the sensitivity change of the optical current sensor.Accordingly, a highly precise Sagnac interferometer-type fiber-opticcurrent sensor can be realized.

Moreover, the signal processing circuit includes the arctangentcompensation means to carry out the compensation to the reference valueor the compensated value, and outputs the compensated value by theabove-mentioned arctangent compensation means as a signal proportionalto the magnitude of the current to be measured.

Therefore, while a normalized output or a compensated normalizationoutput is approximately proportional to the current to be measured inthe area where the current to be measured is small, the arctangent valueof the normalized output or the compensated normalization output isapproximately proportional to the magnitude of the current to bemeasured in the area where the current to be measured is large.Therefore, also even in the area where the current to be measured islarge, it becomes possible to improve the linearity of the input-outputcharacteristic of the Sagnac interferometer-type fiber-optic currentsensor by carrying out the arctangent compensation to the normalizationoutput or the compensated normalization output. As a result, themeasurement accuracy can be raised.

Furthermore, according to this embodiment, the optical filter includesthe first Lyot type depolarizer formed of a polarization maintainingfiber and the polarizer configured using the polarization maintainingfiber optically connected with the phase modulator side. The phasemodulator includes a polarization maintaining fiber to propagate thelight defining the optic axes of the fiber. The polarization maintainingfibers mentioned above, which optically connect between the phasemodulator and the polarizer, constitute the second Lyot typedepolarizer. Moreover, the full length ratio between the first Lyot typedepolarizer and the second Lyot type depolarizer is set 1:2n or 2n:1using the positive integer n.

Therefore, the first Lyot type depolarizer enables to stabilize theoutput light amount from the polarizer. Moreover, since the ratio of thefull length of the first Lyot type depolarizer to the full length of thesecond Lyot type depolarizer is set 1:2n or 2n:1 using the positiveinteger n, it becomes possible to set the group delay time lag of eachdepolarizer beyond a coherent time. Accordingly, it becomes possible tosuppress the interference by the remaining polarization component ineach depolarizer. As a result, it also becomes possible to suppress anoptical phase drift and a zero point drift as the Sagnacinterferometer-type fiber-optic current sensor.

Moreover, the optical filter is equipped with the polarizer and thefirst Lyot type depolarizer that is provided at the light source side ofthe polarizer and formed of the polarization maintaining fiber preparedfor the purpose of specifying the optical axis of the polarizer andpropagating the light.

Therefore, it is not necessary to provide an extra depolarizer byconstituting the first Lyot type depolarizer using the polarizationmaintaining fiber of the polarizer prepared for the purpose ofspecifying an optical axis and propagating light regardless of thefunction of the polarizer. As a consequence, since the number of thepoints in which the respective fibers are connected optically alsodecreases, the characteristics such as optical connection loss, opticaltransmission loss, and a propagation polarization characteristic can bestabilized more, and the optical current sensor becomes more economical.

In addition, an optical crystal element may be used for optical elementssuch as the fiber coupler 103 which is an optical separation means, thedepolarizer 104 b, the polarizer 104 a, and the quarter-wave plate 301in the first embodiment, and it is possible to propagate the light inthe space without using the fiber as the transmitting path (opticalpropagation path). For example, it is possible to use a beam splitterfor the optical separation means, and a Glan-Thompson Prism etc. for thepolarizer 104 a.

In addition, Faraday elements formed of flint glass crystal, fibers orquartz crystal can be also used instead of the sensing fiber 302.

Moreover, in the above, although the third-order harmonics is used amongthe odd-order harmonics as a current corresponding output, otherodd-order harmonics can be used to normalize in the same way. Moreover,in this embodiment, when normalizing the third-order harmonics as thecurrent corresponding output, the second-order harmonics andfourth-order harmonics can be used as even-order harmonics. In addition,the second-order harmonics, fourth-order harmonics and the sixth-orderharmonics are used for compensating the normalized output. Of course,higher even-order harmonics can be used in the same way.

Although the relation between η and k′ is obtained in a range of|θ_(c)≦10° in FIG. 8, if the estimated change of θ_(c) is small, and therelation between η and k′ is obtained in the narrow range, theapproximation accuracy of k′ by η is raised more. Moreover, theapproximation order does not need to be the third-order, and if thehigher order is used to approximate, the approximation accuracy is moreraised.

In addition, although a digital processing circuit is commonly used as apractical circuit composition of the signal processing circuit 109, ananalogue processing circuit can calculate the normalization in analog byusing an analog divider. In case of the compensation, the signalprocessing circuit 109 can calculate the compensation by using an analogmultiplier and an analog adder circuit. Moreover, in case of thearctangent compensation, it is also possible to carry out the arctangentcompensation operation also using the analog multiplier and the analogadder circuit.

2. Second Embodiment

Next, the Sagnac interferometer-type fiber-optic current sensoraccording to the second embodiment is explained with reference to FIGS.13 to 18. In the second embodiment, since the structures other than thephase modulator driving circuit 108 is the same as those of the firstembodiment, the explanation is omitted and the same mark or symbol isdenoted.

The feature of the second embodiment is to set P_(1ω) to “0” at alltimes by controlling the phase modulation depth δ using the phasemodulator driving circuit 108 as shown in FIG. 18.

Next, the control of the phase modulation depth δ to set P_(1ω) to “0”by the phase modulator driving circuit 108 is explained hereinafter. Inthe explanation about the behavior of the light, the Jones matrix isused like the first embodiment.

First, in the above equation 31 showing the first-order harmonics, it isnecessary to set J₁(R) to “0” in order to set P_(1ω) to “0” regardlessof the axial misalignment angle θ_(c) of the optic axis of the mutualpolarization maintaining fibers at Point A in FIG. 1. Here, J₁(R) isdecided by R=2δ sin ω_(m)α. Therefore, ω_(m)α is obtained by dividingthe product of the optical light path length (L_(opt)) specified by thetime lag of time for two circular polarized lights propagating in theFaraday element to receive the phase modulation respectively by thephase modulator and the phase modulation angular frequency (ω_(m)) withthe twice as much velocity of the light (2 c) (namely,ω_(m)α=ω_(m)L_(opt)/2c) according to equation 25. Therefore, if thephase modulation angular frequency ω_(m) is decided, ω_(m)α becomes afixed value (constant value) decided exclusively.

As mentioned above, in the phase modulator driving circuit 108, it isnecessary to control the phase modulation depth δ so as to J₁(R)=0 atall the times in order to set P_(1ω) to “0” at all the times. In otherwords, the phase modulator driving circuit 108 controls the phasemodulation depth δ so that the P_(1ω) becomes “0” by providing a fixedphase modulation depth δ to the phase modulator.

Therefore, it becomes possible to keep the phase modulation actuallyapplied to the light constant by controlling P_(1ω) to “0”, even if thephase modulation efficiency of the phase modulator 105 changes withtemperature or aged deterioration. Therefore, if the phase modulationdepth δ actually applied to the light changes like equations 28 to 36,the output of the Sagnac interferometer-type fiber-optic current sensoralso changes. However, even if the phase modulation efficiency changes,the output change of the Sagnac interferometer-type fiber-optic currentsensor is suppressed by controlling P_(1ω) to “0”.

For the reference, the control to set P_(1ω) to “0” is realized bycontrolling the phase modulation depth δ so as to R=2δ sin ω_(m)α≈3.83.

Furthermore, as above-mentioned, since the phase modulator drivingcircuit 108 controls the modulation so that the P_(1ω) becomes “0”, theorder of odd-order harmonics that can be used as the currentcorresponding output is more than third-order. Since the amplitude ofthe third-order harmonics P_(3ω) is the largest, it becomes possible tomake the influence by the noise small by using P_(3ω) as the currentcorresponding output.

Next, when the third-order harmonics P_(3ω) is controlled so that P_(1ω)becomes “0”, the embodiments to carry out the normalization by dividingthe third-order harmonics P_(3ω) with the amplitudes of the second-orderharmonics, the fourth-order harmonics, and the sixth-order harmonicsusing the normalization means 109 a, and to carry out the arctangentcompensation using the compensation means 109 b of the signal processingcircuit 109 are explained as the first embodiment.

First, the third-order harmonics P_(3ω) when P_(1ω) is controlled to beset to “0” is normalized with |P_(2ω)| |P_(4ω)| or |P_(6ω)| like thefirst embodiment. Thereby, the stable normalized value can be obtained.The normalized value is not affected by the influence of change of thedetected light amount accompanied with the change of the opticaltransmission loss of the optical system and the emitted light of thelight source, and also is not affected by the change of θ_(c) of theSagnac interferometer-type fiber-optic current sensor.

Practically, in the case of θ_(c)=0, the light detected with thedetector follows the equations 15 to 22. Under the condition ofP_(1ω)=0, the current corresponding output P_(3ω)′ which is obtained bynormalizing P_(3ω) with |P_(2ω)| is expressed with the followingequations 56 and 57 as well as the equation 38.

In addition, since the normalization using the fourth-order harmonics orthe sixth-order harmonics is carried out in the same way, theexplanation is omitted. Moreover, the amplitudes of the even-orderharmonics more than eighth-order harmonics is small compared with thesecond-order harmonics, fourth-order harmonics and sixth-orderharmonics, and the harmonics tend to be affected by the influence of thenoise. Furthermore, since the polarization extinction ratio change islarge in the even-order harmonics more than eighth-order harmonics, thepolarity of them may be inverted when the current value to be measuredis large. Therefore, the even-order harmonics more than eighth-orderharmonics are not applicable.

$\begin{matrix}\begin{matrix}{P_{3\omega}^{\prime} = \frac{P_{3\omega}}{P_{2\omega}}} \\{= {{k \cdot \tan}\; \theta_{f}}}\end{matrix} & \begin{matrix}(56) \\(57)\end{matrix}\end{matrix}$

where, k is expressed with the following equation 58.

$\begin{matrix}{k = {\frac{J_{3}(R)}{{J_{2}(R)}} = \left( {\approx 1.044} \right)}} & (58)\end{matrix}$

According to the equation 58, k is a constant obtained exclusivelyregardless of θ_(f) under the condition P_(1ω)=0, and becomes k≈1.044(refer to FIG. 3).

Therefore, θ_(f) is expressed with the following equations 61 and 62 byputting P_(k) with the following equations 59 and 60.

$\begin{matrix}{P_{k} = \frac{P_{3\omega}^{\prime}}{k}} & (59) \\{\approx \frac{P_{3\omega}^{\prime}}{1.044}} & (60) \\{\theta_{f} = {{\frac{1}{4}\arctan \; P_{k}} = {\frac{1}{4}\arctan \frac{P_{3\omega}^{\prime}}{k}}}} & (61) \\{\approx {\frac{1}{4}\arctan \frac{P_{3\omega}^{\prime}}{1.044}}} & (62)\end{matrix}$

As shown in the equations 61 and 62, an output θ_(f) proportional to thecurrent I can be obtained by carrying out the arctangent compensation tothe current corresponding output P_(k) obtained by dividing P_(3ω)′ withthe constant k (refer to the equation 7). In addition, an approximationof θ_(f)≈P_(k)/4 is realized in the area where the current I to bemeasured is small (θ_(f)<<1). That is, the arctangent compensationimproves the measurement accuracy of the Sagnac interferometer-typefiber-optic current sensor in a region where the current I to bemeasured is large.

Next, in the case of θ_(c)≠0, the current corresponding output P_(k)obtained by dividing P_(3ω)′ with the constant k≈1.044 like the case ofθ_(c)=0 follows the equations 29 to 36 as the case of θ_(c)=0. Thecurrent corresponding output P_(k) is expressed with the followingequation 63 by the equations 56 and 59 under the condition of P_(1ω)=0.

$\begin{matrix}\begin{matrix}{P_{k} = \frac{P_{3\omega}}{k{P_{2\omega}}}} \\{= \frac{{{J_{3}(R)}\sin \; 4\theta_{f}} - {2{J_{3}(R)}\sin^{2}\theta_{c}\sin \; 4\theta_{f}}}{k{{{{- {J_{2}(R)}}\cos \; 4\theta_{f}} + \left( {{J_{2}(R)} - {{J_{2}\left( R^{\prime} \right)}\sin^{2}2\theta_{c}\cos^{2}2\theta_{f}}} \right)}}}}\end{matrix} & (63)\end{matrix}$

Since the change ratios of P_(2ω) and P_(3ω) to θ_(c) are comparablehere, good linearity is shown between P_(k) and tan 4θ_(f) (refer toFIG. 13).

As mentioned above, it turns out that it is effective to use the valueobtained by normalizing P_(3ω) with |P_(2ω)| as a current correspondingoutput in order to suppress the deterioration of the linearity withθ_(f), i.e., the measured current.

However, as shown in FIG. 13, in the case of θ_(c)≠0, it turns out thatchange of P_(k) is generated depending on the magnitude of θ_(c)(equivalent to a proportionality coefficient change of P_(k) to tan4θ_(f)). Therefore, even if the arctangent compensation is carried outto P_(k) obtained by the equation 63, error is generated not a little.

In addition, since P_(k) obtained by normalizing P_(3ω) with |P_(2ω)|shows a good linearity with tan 4θ_(f), it is possible to preciselycompensate the output corresponding to the current with a current whichsatisfies 4θ_(f)≦1 at every light current sensing operation afteroptically connecting the fibers at the point A in FIG. 1, if θ_(c) issmall or constant even in the case of θ_(c)≠0. However, suchcompensation is not practical.

Here, since the proportionality coefficient change k′ of P_(k) to tan4θ_(f) varies depending on the magnitude of θ_(c) slightly as shown inFIG. 13 (in the ideal case of θ_(c)=0 and k′=1), tan 4θ_(f), P_(k), andk′ are related like the following equation 64.

tan 4θ_(f) =k′·P _(k)  (64)

Therefore, if the magnitude of θ_(c) is known and change of k′ by θ_(c)is also known, it becomes possible to compensate the equation 64. If theabove equations 32 and 34 are respectively compared with the equation36, the respective changing ratios to θ_(c) can be estimated using avalue obtained by dividing the |P_(2ω)| with |P_(6ω)|.

Here, the attention is paid only about the even-order harmonics becausethe output does not become “0” even in the case of θ_(f)=0 in theeven-order harmonics. In the above case, although the value obtained bydividing |P_(2ω)| with |P_(6ω)| is used to know the magnitude of θ_(c)as a cue, it is also possible to use other even-order harmonics expectzero-order harmonics similarly.

However, in the even-order harmonics more than eighth-order harmonics,the amplitude is small compare with the second-, fourth-, andsixth-order harmonics. Accordingly, it is easier to be affected by theinfluence of the noise. Further, in case the polarization extinctionratio change is large, and the measured current is also large, thepolarity of them may be inverted. As a consequence, the even-orderharmonics more than the eighth-order harmonics are not applicable.

Therefore, the method to estimate θ_(c) using |P_(2ω)| |P_(4ω)| and|P_(6ω)| is superior to raise the measurement accuracy more. If thevalue obtained by dividing |P_(2ω)| with |P_(6ω)| is denoted by η, η canbe expressed with the following equation 65.

$\begin{matrix}{\eta = \frac{P_{2\omega}}{P_{6\omega}}} & (65)\end{matrix}$

Here, η becomes a function of θ_(c) and θ_(f) under the condition ofP_(1ω)=0 as apparent from the above equations 32 and 36. However, ηdefined by the equation 65 hardly changes under the condition that θ_(c)is small or constant with respect to θ_(f) which satisfies 4θ_(f)<<1.Accordingly, η is approximately estimated as a function of θ_(c).

Therefore, if the behavior of θ_(c) with respect to η is evaluated aboutarbitrary θ_(f) which satisfies 4θ_(f)<<1, and η is exclusivelydetermined by θ_(c), the magnitude of θ_(c) can be approximately knownby measuring η. That is, in the case of θ_(f)=0, η is expressed with thefollowing equation 66 under the condition of P_(1ω)=0 from the aboveequations 32, 36, and 65.

$\begin{matrix}{\eta = {\frac{P_{2\omega}}{P_{6\omega}} = \frac{{{- {J_{2}(R)}} + {\left( {{J_{2}(R)} - {J_{2}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}{{{- {J_{6}(R)}} + {\left( {{J_{6}(R)} - {J_{6}\left( R^{\prime} \right)}} \right)\sin^{2}2\theta_{c}}}}}} & (66)\end{matrix}$

Thus, the relation between θ_(c) and η calculated by the equation 66 isshown in FIG. 14. According to FIG. 14, η is decided exclusively byθ_(c) and it becomes possible to get to know the magnitude of θ_(c)approximately by measuring η. Thus, since θ_(c) can be approximatelyknown by η, then the relation between η and k′ related to θ_(c) isevaluated. k′ can be defined as the following equation 64 by the aboveequation 64.

$\begin{matrix}{k^{\prime} = \frac{\tan \; 4\theta_{f}}{P_{k}}} & (67)\end{matrix}$

FIG. 15 shows a result of the evaluation between k′ and η in which k′ isobtained using the equation 67 with respect to the value of θ_(f) whichis not “0” and satisfies 4θ_(f)<<1 under the condition P_(1ω)=0, and ηis also obtained using the equation 65. In this evaluation,4θ_(f)=1.0405E−05 (rad) is used.

As shown in FIG. 15, k′ can be approximately and exclusively decidedusing η defined by the equation 65, and it turns out that thecompensation by the first-order is enough with respect to η. Theevaluated k′ is expressed as the following equation 68 using η.

k′=1.070×10⁻¹·η−8.000×10⁻²  (68)

As described above, under the condition of P_(1ω)=0, if θ_(f) isapproximated by the equations 64 and 68, θ_(f) is expressed with thefollowing equation 69.

$\begin{matrix}{\theta_{f} \approx {\frac{1}{4}{\arctan \left( {\left( {\sum\limits_{n = 0}^{1}{{a_{n} \cdot \eta}\; n}} \right) \cdot P_{k}} \right)}}} & (69)\end{matrix}$

where, a_(n) is a constant, and a₀=−8.000×10⁻² and a₁=1.070×10⁻¹ arepractically used.

Moreover, P_(k) is expressed with the following equation 70, and η isexpressed with above equation 65.

$\begin{matrix}{P_{k} = {\frac{1}{1.044} \cdot \frac{P_{3\omega}}{P_{2\omega}}}} & (70)\end{matrix}$

Therefore, θ_(f) proportional to the measured current value I to beoutputted finally can be obtained based on the equation 69 by usingthree harmonics, P_(2ω), P_(3ω), and P_(6ω). In the case where themisalignment angle of the optic axes between the respective polarizationmaintaining fibers to be optically connected is small or constant (inthe case the value of the polarization extinction ratio is small orconstant), the approximation of θ_(f)≈P_(k)/4 is realized in the areawhere the current I to be measured is small (θ_(f)<<1), and in the areawhere the current I to be measured is large, the arctangent compensationis carried out. Accordingly, the measurement accuracy of the Sagnacinterferometer-type fiber-optic current sensor is improved.

Next, in order to show how much precisely the approximation by the aboveequation 69 is carried out, the relation between θ_(f) and the ratioerror with respect to the misalignment angle θ_(c) between the mutualfibers at point A in FIG. 1 is shown in FIGS. 16 and 17 by defining theratio error by the following equation 71.

$\begin{matrix}{{{Ratio}\mspace{14mu} {error}\mspace{14mu} (\%)} = {\left( \frac{\frac{1}{4}{\arctan \left( {\left( {\sum\limits_{n = 0}^{3}{a_{n} \cdot \eta^{n}}} \right) \cdot P_{k}} \right)}}{\theta_{f}} \right) \times 100}} & (71)\end{matrix}$

As shown in FIGS. 16 and 17, the approximation equation shown by theequation 69 can realize a highly precise approximation, and it becomespossible to approximate the current of θ_(f)<0.06 (rad) with anapproximation accuracy less than 0.1% in the range of θ_(c)=0° to 10°.That is, a highly precise current measurement is realizable bycompensating the output of the optical current sensor using thecompensation means 109 b according to the equation 69.

According to the second embodiment as above, the phase modulation depthis controlled so that the first-order harmonics obtained by thesynchronous detection of the detected light amount with the phasemodulation angular frequency by the detector 106 becomes “0”. Further,the compensation is carried out to the normalized amplitude of thethird-order harmonics with one of the second-, fourth- and sixth-orderharmonics with a value obtained by dividing the amplitude of thesecond-order harmonics with the amplitude of the sixth-order harmonics.Here, even in case where any two of the even-order harmonics of thesecond- to sixth-order harmonics are used, the same effect is obtained.Moreover, the arctangent compensation is carried out to the abovecompensated current to obtain the output of the Sagnacinterferometer-type fiber-optic current sensor. Accordingly, since thecompensated output can be used, even if the polarization extinctionratio of the light transmitting fiber 200 to optically connect thesensor head unit 300 with the signal processing unit 100 deteriorates,it becomes possible to offer the optical current sensor in which thehighly precise current measurement is achieved.

According to this embodiment, since the phase modulation depth of thephase modulator is controlled so that the amplitude of the first-orderharmonics becomes “0”, it becomes possible to carry out the phasemodulation actually applied to the light with a constant modulationvalue, and even if the phase modulation efficiency of the phasemodulator changes with temperature and aging, etc., the output change ofthe Sagnac interferometer-type fiber-optic current sensor can besuppressed.

Furthermore, since the odd-order harmonics are the values approximatelyproportional to the current to be measured, and the odd-order harmonicsand the even-order harmonics are the values approximately proportionalto the detected light amount by the detector. Accordingly, even if it isa case where the detected light amount changes with the emitted lightamount of the light source and the optical transmission loss of theoptical propagation path, etc., a stable output proportional to thecurrent to be measured can be obtained by normalizing the odd-orderharmonics with the even-order harmonics. Especially, since the amplitudeof the first-order harmonics is controlled to become “0”, the third orhigher odd-order harmonics is effective in order to obtain the outputproportional to the current to be measured.

Moreover, since the value of the amplitude of the third-order harmonicsis large as compared with the amplitudes of other odd-harmonics, thethird-order harmonics is hard to be affected by the influence of anoise. Similarly, since the value of the amplitudes of the second-,fourth- and sixth-order harmonics are also large as compared with theamplitudes of other even-order harmonics, the harmonics are also hard tobe affected by the influence of the noise. Therefore, the normalizationoutput obtained by normalizing the amplitude of the third-orderharmonics using one of the amplitudes of the second-, fourth- andsixth-order harmonics shows a good linearity with the magnitude of thecurrent to be measured even if the value of the polarization extinctionratio between the phase modulator and the quarter-wave plate changesunder a condition that the polarization extinction ratio is small orconstant.

In the above embodiment, although the third odd-order harmonics is usedas a current corresponding output, other odd-order harmonics can be usedto normalize in the same way as above. Moreover, although the second-and fourth-order harmonics are used for normalizing the third-orderharmonics as the current corresponding output, and the second-, fourth-and sixth-order harmonics are used to compensate the normalized outputas the even-order harmonics, other even-order harmonics can be used inthe same way as the above.

3. Third Embodiment

Next, the structure of the Sagnac interferometer-type fiber-opticcurrent sensor according to the third embodiment is explained below. Inthe third embodiment, since the components other than the phasemodulator driving circuit 108 is the same as those of the firstembodiment, the explanation about the components shall be omitted andthe same marks or symbols is denoted.

The feature of the third embodiment is that R=2δ sin ω_(m)α is set to belarger than 0 and smaller than 7 in the phase modulator driving circuit108 so that P_(1ω) does not become “0” except R=2δ sin ω_(m)α≈3.83 asexplained later.

Next, the control of the phase modulator by the phase modulator drivingcircuit 108 according to the third embodiment is explained belowincluding the prior problems.

In the second embodiment, the phase modulator driving circuit 108controls the phase modulator so as to be P_(1ω)=0. However, when thecurrent I to be measured is “0”, that is, when θ_(f)=0, P_(1ω) becomes“0” regardless of the magnitude of the phase modulation depth η asapparent from the equation 31. Therefore, the phase modulation η cantake any values, and the control to set P_(1ω)=0 may runaway. Further,if the modulation depth η becomes extremely large, the phase modulator105 may be damaged.

The above unfavorable operation is resulted by the structure of thephase modulator 105. The phase modulator 105 is a Pockels' cell type ora cylindrical piezo-electric tube type configured by twisting an opticalfiber around it. In the both elements, the phase modulation is carriedout by applying a voltage to the Pockels' cell element or a cylindricalpiezo-electric tube. Accordingly, the magnitude of the applied voltageis proportional to the phase modulation depth. In addition, in case ofθ≠0, the conditions to become P_(1ω)=0 exist besides R=2δ sinω_(m)α≈3.83 (refer to FIG. 3).

Therefore, in the third embodiment, the phase modulation depth η ischanged in the range in which the conditions to become P_(1ω)=0 are notsatisfied except R=2δ sin ω_(m)α≈3.83 by the phase modulator drivingcircuit 108 in order to solve such problem. Practically, R=2δ sin ω_(m)αis controlled to be larger than 0 and smaller than 7 in order to avoidto become P_(1ω)=0 except R=2δ sin ω_(m)α≈3.83 as shown in FIG. 3.

Here, ω_(m)α is obtained by dividing the product of the optical lightpath length (L_(opt)) specified by the time lag of time for two circularpolarized lights propagating in the Faraday element to receive the phasemodulation respectively by the phase modulator and the phase modulationangular frequency (ω_(m)) with the twice as much velocity of the light(2 c) (namely, ω_(m) α=ω_(m)L_(opt)/2c) according to the equation 25.Therefore, if the phase modulation angular frequency ω_(m) is decided,ω_(m)α becomes a fixed value (constant value) decided exclusively.

That is, it is possible to control R=2δ sin ω_(m)α≈3.83 by only thephase modulation depth δ. Moreover, the phase modulator driving circuit108 controls R=2δ sin ω_(m)α so as to be larger than 0 and smaller than7. Here, R is obtained by multiply the phase modulation depth δ by asine value of the value obtained by dividing the product of the opticallight path length (L_(opt)) specified by the time lag of time for twocircular polarized lights propagating in the Faraday element to receivethe phase modulation respectively by the phase modulator 105 and thephase modulation angular frequency (ω_(m)) with the twice as muchvelocity of the light (2 c).

Thus, according to the third embodiment, the amplitude of thethird-order harmonics is large, and the condition to control thefirst-order harmonics to become “0” is exclusively decided. Therefore,the control to set the first-order harmonics to “0” can be stabilized.

Moreover, the amplitude of the third-order harmonics becomes smallerunder other conditions than the above in which the first-order harmonicsbecomes “0”. Therefore, the above condition is that in which theamplitude of the third-order harmonics becomes the largest, and thecondition cannot be easily affected by the influence of the noise. Inaddition, in case of the measured current value is “0”, the first-orderharmonics becomes “0”. Thereby, the phase modulation depth of the phasemodulator can take any values, and the control which sets thefirst-order harmonics to “0” may runaway, or if the phase modulationdepth is very large, the phase modulator may be damaged. However, sincea limitation is imposed to the phase modulation depth according to thisembodiment, it also becomes possible to suppress the damage of the phasemodulator.

While certain embodiments have been described, these embodiments havebeen presented by way of example only, and are not intended to limit thescope of the inventions. In practice, the structural elements can bemodified without departing from the spirit of the invention. Variousembodiments can be made by properly combining the structural elementsdisclosed in the embodiments. For example, some structural elements maybe omitted from all the structural elements disclosed in theembodiments. Furthermore, the structural elements in differentembodiments may properly be combined. The accompanying claims and theirequivalents are intended to cover such forms or modifications as wouldfall with the scope and spirit of the invention.

1. A Sagnac interferometer-type fiber-optic sensor, comprising: asynchronous detection circuit to carry out synchronous detection ofdetected light signal with a phase modulation angular frequency of aphase modulator; a signal processing circuit to calculate and output themagnitude of current to be measured using the signal detected in thesynchronous detection circuit; and a phase modulator driving circuit tocontrol the driving of the phase modulator; wherein the phase modulatordriving circuit controls a phase modulation depth of the phase modulatorso that the amplitude of the second-order harmonics and the fourth-orderharmonics of the detected signal obtained by carrying out thesynchronous detection of the detected light signal with the phasemodulation angular frequency becomes the same.
 2. The Sagnacinterferometer-type fiber-optic sensor according to claim 1, wherein thesignal processing circuit includes a normalization means to calculate areference value by dividing the amplitude of the third-order harmonicswith the amplitude of one of the second-, fourth- and sixth-orderharmonics or the sum of the amplitudes of the second- and fourth-orderharmonics, in which the absolute values of the amplitudes of the second-and fourth-order harmonics are controlled to become same by the phasemodulator driving circuit, and the normalized reference value isoutputted as an output signal proportional to the current to bemeasured.
 3. The Sagnac interferometer-type fiber-optic sensor accordingto claim 2, wherein the signal processing circuit includes acompensation means to compensate the reference value with a ratiobetween any two amplitudes of the second-, fourth- and the sixth-orderharmonics, and the compensated value with the compensating means isoutputted as an output signal proportional to the magnitude of thecurrent to be measured.
 4. The Sagnac interferometer-type fiber-opticsensor according to claim 2, wherein the signal processing circuitincludes an arctangent compensation means to carry out the arctangentcompensation to the reference value, and the compensated value by thearctangent compensation means is outputted as an output signalproportional to the magnitude of the current to be measured.
 5. TheSagnac interferometer-type fiber-optic sensor according to claim 3,wherein the signal processing circuit includes an arctangentcompensation means to carry out the arctangent compensation to thecompensated value, and the compensated value by the arctangentcompensation means is outputted as an output signal proportional to themagnitude of the current to be measured.
 6. The Sagnacinterferometer-type fiber-optic sensor according to claim 1 comprisingan optical filter optically connected with the phase modulator forconverting light from a light source to a linear polarized light,wherein the optical filter includes; a first Lyot type depolarizerconstituted by a first polarization maintaining fiber, and a polarizerconstituted by a second polarization maintaining fiber opticallyconnected with the phase modulator side, the phase modulator including athird polarization maintaining fiber to propagate light by defining anoptic axes, wherein a second Lyot type depolarizer is constituted by thesecond and third polarization maintaining fibers to respectively connectbetween the phase modulator and the polarizer, and full length ratio ofthe first Lyot type depolarizer and the second Lyot type depolarizer isset to be 1:2n or 2n:1 using the positive integer n.
 7. The Sagnacinterferometer-type fiber-optic sensor according to claim 1 comprisingan optical filter optically connected with the phase modulator forconverting light from a light source to a linear polarized light, theoptical filter includes; a polarizer, a first Lyot type depolarizerconstituted by a first polarization maintaining fiber provided at thelight source side of the polarizer and to propagate the light bydefining an optic axes of the polarizer, a second polarizationmaintaining fiber provided at the phase modulator side of the polarizerto propagate the light by defining an optic axes of the polarizer, thephase modulator including a third polarization maintaining fiberprovided at the polarizer side to propagate light by defining an opticaxes, and a second Lyot type depolarizer constituted by the second andthird polarization maintaining fibers, wherein the full length ratio ofthe first Lyot type depolarizer and the second Lyot type depolarizer isset to be 1:2n or 2n:1 using the positive integer n.
 8. A Sagnacinterferometer-type fiber-optic sensor, comprising: a synchronousdetection circuit to carry out synchronous detection of the detectedlight signal with a phase modulation angular frequency of a phasemodulator; a signal processing circuit to calculate and output themagnitude of current to be measured using the signal detected in thesynchronous detection circuit; and a phase modulator driving circuit tocontrol the driving of the phase modulator; wherein the phase modulatordriving circuit controls a phase modulation depth so that the amplitudeof the first-order harmonics obtained by carrying out the synchronousdetection of the detected light signal with the phase modulation angularfrequency becomes “0”.
 9. The Sagnac interferometer-type fiber-opticsensor according to claim 8, wherein the signal processing circuitincludes a normalization means to calculate a reference value bydividing the amplitude of the third-order harmonics with the amplitudeof one of the second-, fourth- and sixth-order harmonics, in which theamplitude of the first-order harmonics is controlled to become “0” bythe phase modulator driving circuit, and the normalized reference valueis outputted as an output signal proportional to the current to bemeasured.
 10. The Sagnac interferometer-type fiber-optic sensoraccording to claim 9, wherein the signal processing circuit includes acompensation means to compensate the reference value with a ratiobetween any two amplitudes of the second-, fourth- and the sixth-orderharmonics, and the compensated value with the compensating means isoutputted as an output signal proportional to the magnitude of thecurrent to be measured.
 11. The Sagnac interferometer-type fiber-opticsensor according to claim 9, wherein the signal processing circuitincludes an arctangent compensation means to carry out the arctangentcompensation to the reference value, and the compensated value by thearctangent compensation means is outputted as an output signalproportional to the magnitude of the current to be measured.
 12. TheSagnac interferometer-type fiber-optic sensor according to claim 10,wherein the signal processing circuit includes an arctangentcompensation means to carry out the arctangent compensation to thecompensated value, and the compensated value by the arctangentcompensation means is outputted as an output signal proportional to themagnitude of the current to be measured.
 13. The Sagnacinterferometer-type fiber-optic sensor according to claim 8, wherein thephase modulator driving circuit controls the phase modulation depth sothat the value calculated by the following equation becomes smaller than7 and larger than
 0. $\frac{L_{opt}\omega_{m}\delta}{2c}$ L_(opt):Optical light path length specified by the time lag to respectivelyreceive the phase modulation by the phase modulator δ: Phase modulationdepth ω: Phase modulation angular frequency of the phase modulator c:Light velocity
 14. The Sagnac interferometer-type fiber-optic sensoraccording to claim 8 comprising an optical filter optically connectedwith the phase modulator for converting light from a light source to alinear polarized light, wherein the optical filter includes; a firstLyot type depolarizer constituted by a first polarization maintainingfiber, a polarizer constituted by a second polarization maintainingfiber optically connected with the phase modulator side, the phasemodulator including a third polarization maintaining fiber to propagatelight by defining an optic axes, wherein a second Lyot type depolarizeris constituted by the second and third polarization maintaining fibersto respectively connect between the phase modulator and the polarizer,and full length ratio of the first Lyot type depolarizer and the secondLyot type depolarizer is set to be 1:2n or 2n:1 using the positiveinteger n.
 15. The Sagnac interferometer-type fiber-optic sensoraccording to claim 8 comprising an optical filter optically connectedwith the phase modulator for converting light from a light source to alinear polarized light, the optical filter includes; a polarizer a firstLyot type depolarizer constituted by a first polarization maintainingfiber provided at the light source side of the polarizer and topropagate the light by defining an optic axes of the polarizer and, asecond polarization maintaining fiber provided at the phase modulatorside of the polarizer to propagate the light by defining an optic axesof the polarizer, the phase modulator including a third polarizationmaintaining fiber provided at the polarizer side to propagate light bydefining an optic axes, and a second Lyot type depolarizer constitutedby the second and third polarization maintaining fibers, wherein thefull length ratio of the first Lyot type depolarizer and the second Lyottype depolarizer is set to be 1:2n or 2n:1 using the positive integer n.16. A Sagnac interferometer-type fiber-optic sensor, comprising: asynchronous detection circuit to carry out synchronous detection ofdetected light signal with a phase modulation angular frequency of aphase modulator; a signal processing circuit to calculate and output themagnitude of current to be measured using the signal detected in thesynchronous detection circuit; and a phase modulator driving circuit tocontrol the driving of the phase modulator; wherein the signalprocessing circuit includes; a normalization means to calculate areference value by dividing any one amplitude of the odd-order harmonicsby any one amplitude of the even-order harmonics, in which the harmonicsare obtained by carrying out the synchronous detection of the detectedlight signal with the phase modulation angular frequency, and acompensation means to compensate the normalized reference value with aratio between any two amplitudes of the second-, fourth-, andsixth-order harmonics, and the compensated value by the compensatingmeans is outputted as an output signal proportional to the magnitude ofthe current to be measured.
 17. The Sagnac interferometer-typefiber-optic sensor according to claim 16, wherein, the signal processingcircuit includes an arctangent compensation means to carry out thearctangent compensation to the compensated value, and the compensatedvalue by the arctangent compensation means is outputted as an outputsignal proportional to the magnitude of the current to be measured. 18.The Sagnac interferometer-type fiber-optic sensor according to claim 16comprising an optical filter optically connected with the phasemodulator for converting light from a light source to a linear polarizedlight, wherein the optical filter includes; a first Lyot typedepolarizer constituted by a first polarization maintaining fiber, apolarizer constituted by a second polarization maintaining fiberoptically connected with the phase modulator side, the phase modulatorincluding a third polarization maintaining fiber to propagate light bydefining an optic axes, wherein a second Lyot type depolarizer isconstituted by the second and third polarization maintaining fibers torespectively connect between the phase modulator and the polarizer, andfull length ratio of the first Lyot type depolarizer and the second Lyottype depolarizer is set to be 1:2n or 2n:1 using the positive integer n.19. The Sagnac interferometer-type fiber-optic sensor according to claim16 comprising an optical filter optically connected with the phasemodulator for converting light from a light source to a linear polarizedlight, the optical filter includes; a polarizer a first Lyot typedepolarizer constituted by a first polarization maintaining fiberprovided at the light source side of the polarizer and to propagate thelight by defining an optic axes of the polarizer and, a secondpolarization maintaining fiber provided at the phase modulator side ofthe polarizer to propagate the light by defining an optic axes of thepolarizer, the phase modulator including a third polarizationmaintaining fiber provided at the polarizer side to propagate light bydefining an optic axes, and a second Lyot type depolarizer constitutedby the second and third polarization maintaining fibers, wherein thefull length ratio of the first Lyot type depolarizer and the second Lyottype depolarizer is set to be 1:2n or 2n:1 using the positive integer n.20. A Sagnac interferometer-type fiber-optic sensor, comprising: a lightsource; a fiber coupler optically connected with the light source; asensor head unit including a light sensing fiber formed in a loop and amirror provided an end of the light sensing fiber; a phase modulator tocarry out phase modulation to the light propagating in the light sensingfiber; a phase modulator driving circuit to control the driving of thephase modulator; an optical filter optically connected between the phasemodulator and the fiber coupler; a light photodetector to detect thelight propagating in the light sensing fiber and convert the detectedlight into a detected light signal; a synchronous detection circuit tocarry out synchronous detection of the detected light signal with aphase modulation angular frequency; and a signal processing circuit tocalculate and output the magnitude of current to be measured using thesignal detected in the synchronous detection circuit; and wherein thephase modulator driving circuit controls a phase modulation depth sothat the amplitude of the second-order harmonics and the fourth-orderharmonics obtained by carrying out the synchronous detection of thedetected light signal with the phase modulation angular frequencybecomes the same.
 21. The Sagnac interferometer-type fiber-optic sensoraccording to claim 20, wherein the signal processing circuit includes; anormalization means to calculate a reference value by dividing theamplitude of the third-order harmonics with the amplitude of one of thesecond-, fourth- and sixth-order harmonics or the sum of the amplitudesof the second- and fourth-order harmonics, in which the absolute valuesof the amplitudes of the second- and fourth-order harmonics arecontrolled to become same by the phase modulator driving circuit, and anarctangent compensation means to carry out the arctangent compensationto the normalized reference value, the compensated value by thearctangent compensation means is outputted as an output signalproportional to the magnitude of the current to be measured.
 22. TheSagnac interferometer-type fiber-optic sensor according to claim 20,wherein the sensor head unit includes a quarter-wave plate opticallyconnected to an input end of the sensor head, and the inputted light tothe light sensing fiber is propagated to a clockwise direction and acounterclockwise direction.
 23. The Sagnac interferometer-typefiber-optic sensor according to claim 20, further comprising atransmitting fiber to optically connect the phase modulator and thesensor head unit, wherein the transmitting fiber is once separated, andthe separated transmitting fiber is again optically connected again by alight connector.
 24. The Sagnac interferometer-type fiber-optic sensoraccording to claim 23, the transmitting fiber is formed with thepolarization maintaining fiber.